1) Are there alternative train combinations to get me from Budapest to Mukachevo in a reasonable amount of travel time and in reasonable comfort during daylight hours, and/or back again from Mukachevo to Budapest? I’m willing to accept only one direction during daylight if that is the best combination. I’m willing to go first class if necessary for the best combination.
Actually your task here is really easy, since there is plenty of ways to travel Budapest-Mukachevo during the day. There are 6 trains over the border from Zahony to Chop everyday, so your main task is to take any train from Budapest to Zahony and change there for Zahony-Chop (all the timetables you can find on MAV website -
http://elvira.mav-start.hu/elvira.dll/xsl2/uf). While from Chop to Mukachevo is also no problem - multiple 'suburban' and long-distance trains.
In terms of tickets, the best way to manage everything is to buy one through Budapest-Chop return ticket (valid 1 month) already in Budapest for 34 EUR (it will allow you to make change in Zahony for train to Chop). While in Chop just purchase Ukrainian domestic ticket till Mukachevo, for suburban train it will cost less than 1 USD (app. 6 UAH)
2) Will the Moscow to Budapest train be of a different design or comfort?
Yes, train #15 (called "Tisza") is a sleeper train having seating carriage only on leg Budapest-Chop, but not further. During the night in Chop all the sleeper carriages receive bogie change (in Hungary it is 1435mm, while in Ukraine it is 1512mm), that is why train spends half of the night in Chop. So in order to use on route Budapest-Mukachevo you will need besides ticket (36 EUR) also a reservation for a sleeper berth (up to additional 14 EUR) + you will have 'fun' night with border/customs control and bogie change experience
3) How can I buy a round-trip ticket from Budapest to Mukachevo while I’m still in the U.S.?
We would advice you not to worry about the ticket at all. First of all because there is simply no way to buy it online (such possibility is not offered either by Hungarian or Ukrainian railways) and secondly, there is absolutely no need to do it, because tickets like Budapest-Chop sold in unlimited number and can be used on any trains on the route, thus there is no chance they can be sold out.
p.s. just to give you one of the daylight itineraries: dep. Budapest 10:23 or 10:28 (in Zahony 14:57 and 15:27), catch 15:35 train to Chop, there at 16:52 (account time zone change), all the border/customs things in Chop will take around 20min, then catch 17:52 suburban train to Mukachevo (there 19:11). Departure earlier or later from Budapest also possible.