I think you need to pick up a good guidebook to Peru. You're leaving in less than three weeks and it seems you have a lot of questions unanswered (and a few misconceptions). A guidebook could clear up a lot of this.

Your May 22 day is not possible as you've written it. Your LIM-CUZ flight will not get you to Cusco in time to catch transportation from there to Machu Picchu. Those trains leave very early in the morning. You could catch a train later in the day from Ollantaytambo. That would be your only option if you still want to get to MP that same day, and you'd have to get yourself out there. Check Peru Rail's website for schedules:

www.perurail.com

Through the end of May, the Peru Rail service between Cusco and Ollanta is by bus because of repairs being made to the tracks on that sector. Trains run only between Ollanta and MP for now.

Altitudes:

Cusco = 11,000 ft
Sacred Valley = 9,000 ft
Machu Picchu = 8,000 ft
Aguas Calientes = 6,700 ft

Sure. If you're concerned about altitude, it makes sense to go right to AC/MP or at least out to the Sacred Valley rather than trying to tackle Cusco's lofty heights right away. There are a lot of amazing things to see and do in Cusco, though, so do give it some time toward the end of your trip.

A shuttle bus takes you between Aguas Calientes and Machu Picchu. You could do that by foot as you've indicated, but I think there are better uses of your time.

It's Huayna Picchu, not Huayna Beach.

As for lodging, once again, a guidebook will help you. AC/MP has four VERY expensive hotels in $200-$800 range, and several that are less than $100. Ollanta has a handful of accommodation at a range of prices. There are several lodges around Puerto Maldonado with different prices and comfort levels.

Late May is starting to get into a busy time for tourism in Peru. I'd make some arrangements very soon.