The teapot comparison (to me) seems to be a bad. I got carried away and wrote a wall of text. Feel free to ignore it!

First, lets think about normal probabilities in everyday life. Sometimes there are more ways for one state to come about that another state, for example if I shuffle a deck of cards the number of orderings that look random is much larger than the number of ways (1) of the cards being exactly in order.

However, this manner of thinking only applies to certain kinds of thing - those that are in-principle distinguishable. If you have a deck of blank cards, there is only one possible order, BBBBBB.... To take another example, an electronic bank account might display a total balance of $100. How many different ways are their for that $100 to be "arranged" in that bank account? The same number as 100 coins labelled "1" through "100"? No, of course not. Its just an integer stored on a computer, and their is only one way of picking out the integer 100. The surprising examples of this come from quantum physics, where photons act more like the bank account, where their is only 1 way of a particular mode to contain 100 indistinguishable photons. We don't need to understand the standard model for this, even if we didn't have any quantum theory at all we could still observe these Boson statistics in experiments.

So now, we encounter anthropic arguments like Doomsday. These arguments are essentially positing a distribution, where we take the exact same physical universe and its entire physical history from beginning to end, $U$ (which includes every atom, every synapse firing and so on). We then look at all of the "counting minds" in that universe (people count, ants probably don't, aliens, who knows), and we create a whole slew of "subjective universes", $U_1$ , $U_2$, $U_3$, $U_4$, etc, where each of of them is atomically identical to the original $U$ but "I" am born as a different one of those minds (I think these are sometimes called "centred worlds"). We assume that all of these subjective universes were, in the first place, equally likely, and we start finding it a really weird coincidence that in the one we find ourselves in we are a human (instead of an Ant), or that we are early in history. This is, as I understand it, The Argument. You can phrase it without explicitly mentioning the different  $U$s, by saying "if there are trillions of people in the future, the chances of me being born in the present are very low. So, the fact I was born now should update me away from believing there will be trillions of people in the future". - but the  $U$s are still doing all the work in the background.

The conclusion depends on treating all those different subscripted  $U$s as distinguishable, like we would for cards that had symbols printed on them. But, if all the cards in the deck are identical there is only one sequence possible. I believe that all of the $U_1$ , $U_2$, $U_3$, $U_4$'s etc are identical in this manner. By assumption they are atomically identical at all times in history, they differ only by which one of the thinking apes gets assigned the arbitrary label "me" - which isn't physically represented in any particle. You think they look different, and if we accept that we can indeed make these arguments, but if you think they are merely different descriptions of the same exact thing then the Doomsday argument no longer makes sense, and possibly some other anthropic arguments also fall apart. I don't think they do look different, if every "I" in the universe suddenly swapped places - but leaving all memories and personality behind in the physical synapses etc, then, how would I even know it? I would be a cyborg fighting in WWXIV and would have no memories of ever being some puny human typing on a web forum in the 21s Cent. Instead of imaging that I was born as someone else I could imagine that I could wake up as someone else, and in any case I wouldn't know any different.

So, at least to me, it looks like the anthropic arguments are advancing the idea of this orbital teapot (the different scripted  $U$s, although it is, in fairness, a very conceptually plausible teapot). There are, to me, three possible responses:

1 - This set of different worlds doesn't logically exist. You could push this for this response by arguing "I couldn't have been anyone but me, by definition." [Reject the premise entirely - there is no teapot]

2 - This set of different worlds does logically make sense, and after accepting it I see that it is a suspicious coincidence I am so early in history and I should worry about that. [accept the argument - there is a ceramic teapot orbiting Mars]

3 - This set of different worlds does logically make sense, but they should be treated like indistinguishable particles, blank playing cards or bank balances. [accept the core premise, but question its details in a way that rejects the conclusion - there is a teapot, but its chocolate, not ceramic.].

So, my point (after all that, Sorry!) is that I don't see any reason why (2) is more convincing that (3).

[For me personally, I don't like (1) because I think it does badly in cases where I get replicated in the future (eg sleeping beauty problems, or mind uploads or whatever). I reject (2) because the end result of accepting it is that I can infer information through evidence that is not causally linked to the information I gain (eg. I discover that the historical human population was much bigger than previously reported, and as a result I conclude the apocalypse is further in the future than I previously supposed). This leads me to thinking (3) seems right-ish, although I readily admit to being unsure about all this.].

"Random" is the null value we can give as an answer to the question "What is our prior?"

I think the word you are looking for here is "equiprobable".

It's propper to have equiprobable prior between outcomes of a probability experiment, if you do not have any reason to expect that one is more likely than the other.

It's ridiculous to have equiprobable prior between states that are not even possible outcomes of the experiment, to the best of your knowledge. 

You are not an incorporeal ghost that could've inhabited any body throughout human history. You are your parents child. You couldn't have been born before them or after they are already dead. Thinking otherwise is as silly as throwing a 6 sided die and then expecting to receive any outcome from a 20 sided die.

I was anthropically sampled out of some space

You were not anthropically sampled. You were born as a result of a physical process in a real world that you are trying to approximate as a probability experiment. This process had nothing to do with selecting universes that support conscious processes. This process has already been instantiated in a specific universe and has very limited time frame for your existence.

You will have to ignore all this knowledge and pretend that the process is completely different, without any evidence to back it up, to satisfy the conditions of Doomsday argument. 

I am more so rejecting the magical results that people take away from anthropic reasoning, especially when people use incorrect assumptions, and I'm basically rejecting anthropic reasoning as something that is irredeemably weird or otherwise violates Bayes.

To me the more interesting thing is not the mechanism you must invent to protect Heisenberg uncertainty from this hypothetical classical gravitational field, but the reason you must invent it. What, in Heisenberg uncertainty, are you protecting?

Roughly speaking, it's the ability for a quantum atom that goes through a double slit experiment to exhibit superposition of different locations, and it creates an interference pattern when it reaches the detector, which at this point is well established by experiments, and if you don't want to quantize gravity, ala quantum gravity approaches, but still want to reproduce quantum behavior, then you need to introduce physically random stuff.

Does standard QED, by itself, contain something of the in-the-territory state-of-omitted-knowledge class you imagine precludes anthropic thinking?

No, because it's not a theory that includes gravity.

If not, what does it contain, that requires such in-the-territory obscurity to preserve its nature when it comes into contact with a force field that is deterministic?

The uncertainty principle fundamentally is inconsistent with a classical, deterministic field.

If we assume that gravity is quantum in our universe, which is likely but not certain, then there's no problem, but in quantum universes with classical gravity, then we need noise to make both theories consistent with each other.

The classical part was important here.

Good question! I didn't actually think about this consciously, but I guess my intuitive assumption is that the sufficiently advanced civilizations are strongly constrained by the laws of physics, which are the same for everyone, regardless of their intelligence.

A genius human inventor living in 21st century could possibly invent a car that is 10x faster than any other car invented so far, or maybe a rocket that is 100x faster than any other rocket. But if an alien civilization already has spaceships that fly at 0.9c, the best their super-minds can do is to increase it to 0.99c, or maybe 0.999c, but even if they get to 0.999999c, it won't make much of a difference if two civilizations on the opposite sides of the galaxy send their bombs at each other.

Similarly, human military in 21st century could invent more powerful bombs, and then maybe better bunkers, and then even more powerful bombs, etc. So the more intelligent side can keep an advantage. But if the alien civilizations invent bombs that can blow up stars, or create black holes and throw them at your solar system, there is probably not much you can do about it. Especially, if they won't only launch the bombs at you, but also at all solar systems around you. Suppose you survive the attack, but all stars within 1000 light years around you are destroyed. How will your civilization advance now? You need energy, and there is a limit on how much energy you can extract from a star; and if you have no more stars, you are out of energy.

Once you get the "theory of everything" and develop technology to exploit nature at that level, there is nowhere further to go. The speed of light is finite. The matter in your light solar system is finite; the amount of energy you can extract from it is finite. If you get e.g. to 1% of the fundamental limits, it means that no invention ever can make you 100x more efficient than you are now. Which means that a civilization that starts with 100x more resources (because they started expanding through the universe earlier, or sacrificed more to become faster) will crush you.

This is not a proof, one could possibly argue that the smarter civilization would e.g. try to escape instead of fighting, or that there must be a way to overcome what seems like the fundamental limitations of physics, like maybe even create your own parallel universe and escape there. But, this is my intuition about how things would work on the galactic scale. If someone throws a sufficient amount of black holes at you, or strips bare all the resources around you, it's game over even for the Space Einstein.