What are efficient Dyson spheres probably made of?

There are many possible Dyson sphere designs, but they seem to fall into three broad categories: shells, orbital swarms, and bubbles. Solid shells are probably unrealistic. Known materials aren't strong enough. Orbital swarms are more realistic but suffer from some problems with self-occlusion and possibly collisions between modules. Limitations on available materials might still make this the best option, at least at first.

But efficient Dyson spheres are probably bubbles. Rather than being made of satellites, they're made of statites, that is, solar sails that don't orbit, but hover. Since both gravitational acceleration and radiant intensity follow the inverse square law, the same design would function at almost any altitude above the Sun, with some caveats. These could be packed much more closely together than the satellites of orbital swarms while maybe using less material. Eric Drexler proposed 100 nm thick aluminum films with some amount of supporting tensile structure. Something like that could be held open by spinning, even with no material compressive structure. Think about a dancer's dress being held open while pirouetting and you get the idea.

The radiation needs to be mostly reflected downwards for the sails to hover, but it could still be focused on targets as long as the net forces keep the statites in place. Clever designs could probably approach 100% coverage.

What percent of the solar system can be converted into Dyson-sphere material? Are gas giants harvestable?

Eventually, almost all of it, but you don't need to to get full coverage. Yes, they're harvestable; at the energy scales we're talking about, even stellar material is harvestable via star lifting. The Sun contains over 99% of the mass of the Solar System.

How long would it take to harvest that material?

I don't know, but I'll go with the 31 years and 85 days for an orbital swarm as a reasonable ballpark. Bubbles are a different design and may take even less material, but either way, we're talking about exponential growth in energy output that can be applied to the construction. At some point the energy matters more than the matter.

What would the radius of a Dyson sphere be? (i.e. how far away from the sun is optimal). How thick?

I'd say as close to the Sun as the materials can withstand (because this takes less material), so probably well within the orbit of Mercury. Too much radiation and the modules would burn up. Station keeping becomes more difficult when you have to deal with variable Solar wind and coronal mass ejections, and these problems are more severe closer in.

The individual statite sails would be very thin. Maybe on the order of 100 nm for the material, although the tensile supports could be much thicker. I don't know how many sails an optimal statite module would use (maybe just 1). But the configuration required for focus and station keeping probably isn't perfectly flat, so a minimal bounding box around a module could be much thicker still.

An energy efficient Dyson Sphere probably looks like a Matrioshka brain, with outer layers collecting waste heat from the inner layers. Layers could be much farther apart than the size of individual modules.

If the sphere is (presumably) lots of small modules, how far apart are they?

Statites could theoretically be almost touching, especially with active station keeping, which is probably necessary anyway. What's going to move them? Solar wind variation? Micrometor collisions? Gravitational interactions with other celestial bodies? Remember, statites work about the same regardless of altitude, so there can be layers with some amount of overlap.

"if an AI is moderately 'nice', leaves Earth alone but does end up converting the rest of the solar system into a Dyson sphere, how fucked is Earth?

Very, probably. And we wouldn't have to wait for the whole (non-Sun) Solar System to be converted before we're in serious trouble.

The planet Mercury is a pretty good source of material:

Mass: $3.29\times {10}^{23}$ kg (which is about 70% iron)

Radius: $2.44\times {10}^6$ m

Volume: $6.08\times {10}^{19}$ m^3

Density: $5411$ kg/m^3 

Orbital radius:  $0.39AU=5.79\times {10}^{10}$ m

A spherical shell around the sun at roughly same radius as Mercury's orbit would have a surface area of $4.21\times {10}^{22}$ m^2, and spreading out Mercury's volume over this area gives a thickness of about 1.4 mm. This means Mercury alone provides ample material for collecting all of the Sun's energy via reflecting light – very thin spinning sheets could act as a swarm of orbiting reflectors that focus sunlight onto large power plants or mirrors that direct it to elsewhere in the solar system. Spinning sheets could be made somewhere between 1-100 μm thick, with thicker cables or supports for additional strength, perhaps 1-10 km wide, and navigate using radiation pressure (using cables that bend the sheet, perhaps). Something like ${10}^{15}$ or ${10}^{16}$ mirrors would be enough to intercept and redirect all of the sun's light.

The gravitational binding energy of Mercury is on the order of ${10}^{30}$ J, or on the order of an hour of the Sun's output. This means in theory the time it takes for a new mirror to pay it's own manufacturing energy cost is in principle quite small; if each kg of material from Mercury is enough to make on the order of 1-100 square meters of mirror, then it will pay for itself in somewhere between minutes and hours (there are roughly 10,000 w/m^2 of solar energy at Mercury's orbit, and each kg of material on average requires on the order of ${10}^7$ J to remove). Only 40-80 doublings are required to consume the whole planet depending on how thick the mirrors are and how much material is used to start the process. Even with many orders of magnitude of overhead to account for inefficiency and heat dissipation, I believe Mercury could be disassembled to cover the entire sun with reflectors on the order of years and perhaps as quickly as months; certainly within decades.

Ooh boy this is a fun question:

For temperature reasons, a complete Dyson sphere is likely to be built outside the earth, as the energy output of the sun would force one at 1 A.U. to be 393K = 119 C. I assume the AI would prefer not to run all of its components this hot. A sphere like that would cook us like an oven unless the heat dissipating systems somehow don't radiate any energy back inwards (which is probably impossible).

A Dyson swarm might well be built at a mixture of inside and outside the earth's orbit. In that case the best candidate is to disassemble mercury, using solar energy to power electrolysis to turn the crust into metals, send up satellites to catch more sunlight, and focus that back down to the surface.

Mercury orbits at 60 million km from the sun. This means a circumference of 360 million km. The sun is 1.2 million km across, but because it's at 0.38 au from the sun, a band which blocks out the sun for the earth entirely would only need to be 0.8 million km. This gives a total surface area of 290e12 square kilometers to block out the sun entirely. Something like a Dyson belt.

If the belt is 1 m thick on average, this gives it a total volume of 290e18 cubic meters. Mercury has a volume of 60 billion cubic km = 60e18 cubic meters. This would blot out approximately 1/5 of the sun's radiation.

To put things in perspective, Mars is kinda maybe almost habitable with a lot of effort and gets less than 1/2 of the sun's radiation. I would make a wild guess that with 80% of the solar radiation we could scrape by with immense casualties due to massive decreases in agricultural yield. Temperature is somewhat tractable due to our ability to pump a bunch of sulfur hexafluoride into the atmosphere to heat things up.

As a caveat, I would suggest that if the AI is "nice" enough to spare Earth, it's likely to be nice enough to beam some reconstituted sunlight over to us. A priori I would say the niceness window for "unwilling to murder us while on earth, and we pose a direct threat, but unwilling to suffer the trivial cost of keeping the lights on" is extremely narrow.