Their evaluation & explanation sounds legitimate. I would assume that when the coffee was spilled it made it's way into the keyboard. If this is the case it's entirely reasonable to assume that it would make it's way through the keys and inside the top case, with the next component it reaches being the logic board.
In regards to your two questions:
1) Without seeing the Mac and only relying on my experience (see below) I can only make an assumption, but from the information given the list of parts does seem very likely. Yes, the logic board contains a majority of components for your Mac. The processor, GPU, SMC, some ports, etc are all located on this one major component. With the way I'm assuming the spill occurred the coffee wouldn't have made it to the SSD, which connects directly to the logic board, thanks to the SSD being located on the opposite side of the logic board. The only fault I see with their explanation is the speakers, I haven't replaced a Retina top case in a while but I could've sworn that the speakers are still separate components.
2) There's no way for them to tell you upfront if the parts are new or refurbished. Apple re-uses eligible (meaning not beyond economical repair) parts after a strict refurbishing process that includes individual testing of components. When technicians receive parts there is no distinguishing feature to know if it's used or newly manufactured. Honestly, the only way to be sure is if you request a part for a Mac that was just released, simply because there wouldn't have been time to collect used parts and put them through the refurbishing cycle. Either way in my experience I've never seen a trend of service (repair) parts being faulty. It DOES happen from time to time but I've never seen a trend of obviously refurbished parts leading to repeat repairs or premature failures. Apple's replacement parts do carry a 90 day warranty or take over the rest of your existing AppleCare warranty, whichever is a longer period.
Did the AASP mention a flat rate or depot repair to you? I haven't priced out the parts they mentioned in a while b considering the amount of damage that might be a little bit cheaper. Basically Apple offers tiered flat rate repairs for out of warranty situations. Since this is liquid damage it would likely be a tier 4, the highest of them, but it might still be a little cheaper. Might be something to ask them about.
Source - I worked as a Genius for several years, as an in-store and onsite AppleCare certified technician for several years and have dealt with AppleCare extensively on both sides (as a technician and as a representative for an enterprise customer) for several years. I've repaired well over 1000 MacBooks, around 1000 iMacs, several hundred MacBook Pros/Airs and a couple hundred Mac minis and Mac Pros. Everything from in warranty failures to severe fire and liquid damage.
Best Answer
Restarting your machine won't damage it physically. The only damage that you can really do to your machine is to the software and/or firmware. The only way that restarting can do that type of damage is if you restart while operations are still active (i.e. an EFI update or disk activity). Other than that, you are pretty safe.