Will #2 copper THHN be sufficient cable to run to the subpanel?

At that distance I would run aluminum. aluminum feeder is fine for most things and significantly less expensive.

Your load is maxed at 95 amps with #2 copper but going to aluminum we usually jump a size or 2.

at 1 awg aluminum you are good for 100 amps, below 100 amps / 1awg code requires the 60 degree table to be used.

With #1 aluminum at 100 amps the voltage drop will only be 1.86% so you would be good to go all the way around for a full 100 amp sub.

The 2020 code requires a disconnect at the meter but there are many states not on the 2020 code yet if ever.

A few notes first

As you probably know, North American power is split-phase - it is 240V with neutral in the middle, giving 2 opposite poles or legs of 120V. So if this has a 50A breaker, that means you have 120V @ 50A on one leg, and 120V @ 50A on the other leg.

So it's 50A worth of 240V... but 100A worth of 120V.

By the way, I assume you have one breaker which is double the normal width of a breaker. It looks a lot like 2 breakers whose handles are tied, but it is in fact one breaker, and it has an internal "common trip" mechanism which is required for this application. So it must be 1 breaker, not 2 sandwiched.

Voltage drop and wire selection

The max you're allowed to plan to use is 80% of breaker trip. So with a 50A breaker you could only plan to use 40A. Your needs are much lower than that, given that, after all, we're talking two legs of 120V @ 40A each. So we'll compute on 40A.

Calculating for 40A @ 240V, using #6 copper wire gives 2.21% voltage drop. #4 aluminum wire gives 2.29% voltage drop. Both are extremely acceptable; even the wire salesman considers these really good numbers.

Honestly, I don't normally even bother doing the calculations until it's farther than 180 feet.

Pro electricians wiring their own houses would use aluminum. There is some scary talk about aluminum out there, but it applied to a mistake in the 1960s trying to downscale power-line alloy aluminum for small circuit use (15-20A) relating to how the wire was landed on lugs (screw terminals and backstabs). Aluminum terminations have to be managed more carefully, but doing that safely is super easy with large feeder like this. Even 1960s-era aluminum feeder is considered safe; the new stuff is awesome.

Anyway, the panel lugs themselves are aluminum, because aluminum lugs are the universal donor - they play well with both copper and aluminum wire. (that gives a hint as to the source of the above problem).

#2 copper would be insane overkill. Do not buy #2 copper. If you already have a 450' coil of #2 copper sitting around, sell it on eBay and use some of the profits to buy all the wire you'd need. Pocket the rest.

The #4 aluminum I recommend is about 1/7 the cost of #2 copper, e.g. the aluminum is $150 US vs $1000+.

If you really want to spend that much on wire, #2 copper is a bad choice because you are splitting the switch. It's far too big for your "few outlets" and far too small to carry the full 200A service from garage to house. If you ever reversed flow so the service came into the garage, you'd have to replace all that #2.

If your aim is to put in "the ultimate" wire once, and never trench again, well we recommend conduit for the "never trench again" part... but that aside, I would lay 250 kcmil aluminum. It's good for 225A service, will handle 200A without worry for voltage drop issues, and it's still much less than #2 copper.

Not being inappropriately scared of aluminum wire will save you money. For large size feeders, aluminum is safe and dependable - in fact, the lugs the wires connect to are aluminum. The "scary aluminum wiring" of yore is a particular alloy you will not normally find in wiring today, and mostly a problem with 15 and 20A house wiring, not large feeder circuits.

Paying attention to your proposed feed amperage (50A) and your loads (as described, a lot less than 50A) the size of the panel you are feeding (200A) does not come into play for sizing this wire.

The feed breaker amperage does set a minimum wire size. The distance would come into play if you were taxing that feed breaker, but two door openers and some hand tools won't load it much at all, and unless you are lighting a stadium the LED lights will hardly register.

Choosing conservatively from the 60C column since I don't know how you propose to wire this, 4AWG aluminum would be perfectly adequate for 50A service, and will even have low voltage drop even if fully loaded.