I've been to a number of Austrian and Swiss ski places so I have some direct experience.
1) Lech is great and you can get to St. Anton for some skiing also. Fantastic downhill and cross country. Plenty of red and black and off piste. We stayed in St. Anton once and in Lech once and had a great time in both places. Also consider Galtur / Ischgl which is really (really!!) extensive and great (quieter to stay in Galtur). A train will get you to St. Anton but you'll need to take the bus or a taxi to get to Lech. We got a car service from INN to Lech and once to Galtur - much easier if a bit more costly. For your follow on trip you could take the bus from Lech to St Anton and then by train wherever.
I'd say you can get to Lech from Innsbruck the day you arrive and rest there unless you have some reason to stay in Innsbruck (nice but not much and a bit touristy for my taste). I prefer to get where I'm going and 'recover' there, but don't go by me: I flew to ZRH, took the train to Davos and skied half a day that same day!
2) Early or mid march is a good time to go, based on my experience..
3) Booking.com has worked well for me - I scored a single in a barely-three-star-but-fine-for-me place last March in Davos that was <$50/night with breakfast. Lots of availability but worth checking the town website/tourist info eg for
https://www.lechzuers.com/en
4) I always bring helmet and boots and nothing else. I rented skis and poles in Davos from the place right at the lift and it was $180 for 5 days for the middle-tier package. Austria will be less and it's great not lugging skis around on trains.
5) Just MHO but get used to the idea of walking a couple hundred meters in ski boots or have some lightweight packable snow boots. Also there's almost always a bus that will get you close to where you are going.