Electrical – install a 60A main breaker in a load center rated for 125A

This sounds good overall, with some exceptions. (Also, I basically just set up something similar very recently).

Physical: Sounds good. 90' isn't too bad, and not something where you should have to worry about more than a couple percentage points voltage drop.

Subpanel: If you are using a 100A subpanel, why swap the breaker for a 60A breaker? I am guessing this panel is fed by a breaker in your main panel so that breaker would be rated to protect whatever wire is run. For example, if you run 2AWG Al, it can be protected by a 100A breaker. You also don't need a main breaker in your shed's panel, but it might be nice for convenience.

Wire: I'm not sure what's available to you or what anything costs where you are, but 2AWG is very large, and a 4 wire cable of 2-2-2-4 would be pretty ridiculously stiff. Why not run 4AWG Cu THHN/THWN, which will be easier to work with, and potentially cost less? I'm also not quite sure what you mean about upgrading to 90A based on a 75°C rating... Even running 2-2-2-4 you can get a full 100A without needing to do anything. See below for more about grounding.

Answers:

• Your wire choice seems fine, but I'm still confused about your breaker choice
• THWN is the insulation that allows wet rating. The NEC also specifies several other cables that are suitable for wet locations, including MTW, RHW, RHW-2, TW, THW, THW-2, THHW, THWN, THWN-2, XHHW, XHHW-2, ZW
• Running cable through conduit can be difficult when it's large cable. It really depends on the number of bends, wire gauge, and what equipment you have available
• Colors matter but you can wrap the ends with the appropriate colors to ensure you know which is which
• Check out this question if you haven't already seen it. It has a great answer for what wires you should pull (including an EGC).

The answer is "electrician in trade since 1976". In 1976 a 3-wire groundless connection was legal. Certainly not today.

Even a 1976, a separate neutral and ground wire was an improvement. He made it sound like the AHJ would prohibit it, seems unlikely.

Today, belt and suspenders is mandatory. All sub panels must have ground and neutral brought separately, with separation in the sub panel. And then, an outbuilding or other off site location needs a separate ground rod.

And unless the ground rod tests to 25 ohms, it needs two ground rods. Unless you have the equipment, testing is more expensive than driving a second ground rod.

If you're in conduit, use THWN or XHHW. It's cheaper.

if the conduit run is continuous, don't choke cable down it, use individual THWN or XHHW conductors. Your neutral can be smaller and your ground can be much smaller.

For instance I just priced 2/2/4/6 in XHHW and it was $120 for 100 feet. I don't see any 2/2/2/4 URD or anything out there at that price.

Mind you I got these prices by calling my local electrical supply house: trying to buy wiring on the Internet doesn't really work. Too many prices are sucker prices, or lowball but "gee, we're out of stock but thanks for joining my email list" prices, and then there's shipping to contend with. There is also junk imported from far-east places that use US wire sizes. You don't want that. US big-box stores don't have any selection. Proper electrical supply houses aren't online, these things have a very low price-to-weight ratio, so selling online is a fool's game.

Why URD is not a good fit for feeder

Feeder, service lateral and distribution are three separate and different things. Your location is on your side of the meter, which makes yours feeder.

Power companies follow a completely different and more liberal code. They are allowed to use AA-1350 aluminum, it was designed for power transmission and distribution after all! URD is intended for distribution (pole to pole) or service lateral (pole to demarcation point which is before the meter) and that is abundantly clear:

• from product descriptions (which say that plainly, even if many don't understand the terms and so, disregard them)
• product brochures (note the truly screwball ampacity ratings, and footnote that "by the way" these don't apply to NEC usage which must come from NEC's tables). This is glossed over because product users are expected to know it.

Now if you selected this cable because of these wild ampacity ratings like 120A for #4, that doesn't work at all. You need to use NEC rules for determining ampacity. For instance #4 Al has the same ampacity as #6 Cu, only good for a 60A subpanel.

So why does groundless 2/2/2/4 URD exist? For service laterals for three-phase wye service, hot-hot-hot-neutral, either wye or wild-leg delta. Why does Home Depot sell it? Because it sells. Given how bad their help is, I'm sure a lot of what they sell is misapplied, or wasted when the inspector catches it.

Also, the cable does not have a ground wire. People look at the yellow striped smaller wire and go "that's the ground because yellow is kinda like green". No, it's the neutral. It's legal to remark #4 or larger wires, so it would need to be marked with green tape to make it a ground, and a hot conductor would need to be re-marked with white tape to be neutral. I bet almost nobody does that.

A fair bit of googling reveals many claims that URD is not legal inside any home. The claims are that NEC doesn't mention URD so it's not a proper wiring method. That does make sense.