Q2. The train operator is Avanti West Coast, so book through them- https://www.avantiwestcoast.co.uk/
Q1 . Personally I think that AWC 1st class is over priced, vis a vis standard class. Instead pay an extra £35 at the time of booking or on train for Standard Premium- https://www.avantiwestcoast.co.uk/travel-information/onboard/standard-premium
This is first class without the free food and drink or lounge access.
Standard class is very adequate. What I always suggest, if you get an 11 car train, is taking a seat in the un-reserved standard class cars U and G (the 2 rear standard class cars) as those are the quietest ones. I always do that, even if I have a seat booked elsewhere on the train. If it is a 9 car set take your allocated seat. 9 or 11 cars can't be predicted ahead of time.
A few weeks ago I actually started to board an 11 car train at London Euston, when it was suddenly switched out to a 9 car set. The 11 car one was re-allocated to another route. In that case I had been too clever. Along with 40 or 50 other people I knew the platform ahead of it being advertised, had checked the destination displays on the train and was pre or priority boarding. So my own fault. But even the train crew were taken by surprise.
Q3. You can book from 12 weeks out and should do so for the very best prices. But don't sweat it, I usually book about 2 to 3 (maybe 4 at a push) weeks out and still get very, very keen prices. Booking well ahead fares start at £35 or £70 in Standard Premium.
Q4. Yes you can book until the moment of departure. For next Tuesday Advance fares start at £80 (or £115 standard premium), the walk up flexible fare is £125.
Booking on AWC direct you also get to choose your seat from a plan.
EDIT- I note that you say this is your only trip. So if you get the tickets at the lead in price (or anything like it) and do not opt for Premium Standard a railcard will not be worthwhile for you.