I have no idea whether I can post anything here yet but I'll try to answer: all other arguments other than the one from basic chemistry are irrelevant. organic life may be easily construed as a controlled combustion of organic fuel, one drawn over a very long span as opposed to happening all at once like an actual fire, and that is a chemical reaction that proceeds in the direction of increasing entropy: all organic foodstuffs are in the condensed phase (liquids and solids) and their combustion processes are gases. Ergo, organic life increases entropy over all. Simple, right? ~Chara 

I think one shouldn't think of entropy as fundamentally preferred or fundamentally associated with a particular process. Note that it isn't even a well-defined parameter unless you posit some macrostate information and define entropy as a property of a system + the information we have about it.

In particular, life can either increase or decrease appropriate local measurements of entropy. We can burn the hydrocarbons or decay the uranium to increase entropy or we can locally decrease entropy by changing reflectivity properties of earth's atmosphere, etc.

The more fundamental statement, as jessicata explains, is that life uses engines. Engines are trying to locally produce energy that does work rather than just heat, i.e., that has lower entropy compared to what one would expect from a black body. This means that they have to use free energy, which corresponds tapping into aspects of the surrounding environment where entropy has not yet been maximized (i.e., which are fundamentally thermodynamic rather than thermostatic), and they also have to generate work which is not just heat (i.e., they can't just locally maximize the entropy). Life on earth mostly does this by using the fact that solar radiation is much higher-frequency than black-body radiation associated to temperatures on Earth, thus contains free energy (that can be released by breaking it down).

1. We do not expect increasing entropy a priori, because Second Law is true only in closed systems. Open systems in general case have arbitrary entropy production. Under some nice conditions, Prigogine's theorem shows that in open systems entropy production is minimal. And the Earth, thanks to the Sun, is open system.
2. You analyze wrong components of life. The main low entropy components are membranes, active transport, excretory system, ionic gradients, constant acidity levels, etc. Oxygen is far down the list, because oxygen is actually a toxic waste from photosynthesis.

I kinda think of 'free energy' and 'entropy' as being things that living creatures in some sense 'consume'. We use the 'order' present in the universe to advance our goals (e.g. homeostasis) and leave behind a trail of higher entropy. We harness a gradient of incoming energy and order.

The sunlight which a plant absorbs might counterfactually have been turned into heat after being absorbed by the ground, and ended up in the same entropy state (from the perspective of the universe). Or it might have reflected, traveled light-years through space, and warmed some other thing. The leaf managed to insert itself in this process, intercepting the free energy, and more rapidly-than-counterfactually-expected increased the entropy of the universe.

And living multi-cellular beings are basically made up of tiny entities, cells, which are generally doing the metabolism process internally. And then mitochondria and chloroplasts within cells. But it would be a mistake to say that the living thing is causing itself to be disordered because it's increasing entropy in parts of itself. It's spending free energy (and 'excreting' entropy) in order to accomplish things. For instance, using a muscle (converting some of its stored energy to motion and waste heat) in order to bring food to the creature's mouth. The creature is creating an anti-entropic state, pursuing its specific goals, by increasing the probability of the universe corresponding to its goal state, by causing other things to be extra entropic (always with some extra loss along the way, like from friction). You are missing the order that the agent is creating in the world though if you aren't analyzing the world with the frame of how likely the agent's goals were to be achieved by random chance (e.g. Brownian motion) versus by active optimization efforts by the agent. Anytime a living creature agentically does anything, they are consuming free energy and excreting entropy.

That's my understanding anyway, but I may be using the physics terms wrong since I'm not a physicist.

The amount of entropy in a given organism stays about the same, though I guess you could argue it increases as the organism grows in size. Reason: The organism isn't mutating over time to become made of increasingly high entropy stuff, nor is it heating up. The entropy has to stay within an upper and lower bound. So over time the organism will increase entropy external to itself, while the internal entropy doesn't change very much, maybe just fluctuates within the bounds a bit.

It's probably better to talk about entropy per unit mass, rather than entropy density. Though mass conservation isn't an exact physical law, it's approximately true for the kinds to stuff that usually happens on Earth. Whereas volume isn't even approximately conserved. And in those terms, 1kg of gas should have more entropy than 1kg of condensed matter.

I think the confusion may arise from this concept of 'entropy density'?

To compare density levels, we need to look at a specific volume or amount of matter (recall the units of entropy are energy over temperature, no reference to space or mass). In this closed system the 2nd law tells us that overall entropy only goes up, but it does not help us to differentiate between different areas of density. It also tells us that, overall, the density will increase over time, which is not intuitive.

Considering open systems makes things easier. Energy and matter can flow in and out. You can still sample your 'entropy density' in defined volumes, and you may indeed find that 'life has increased entropy locally'. But, through thermodynamic coupling, a greater amount of entropy has been exported to the environment. This coupling is the 'life engines' Dimitry and jessicata refer to.

In summary, the 'entropy density' concept needs to be considered carefully in local vs. global terms.