T-Mobile unlock policy requires you to have had service with them for at least 40 days (60 if prepaid) and have not requested an unlock code within the last three months. If you paid full price for the phone at a T-Mobile store or authorized dealer you can unlock whenever you want but you will have to fax in your proof of purchase.
The problem with getting the 820 is that, if you unlock it and bring it to T-Mobile after your contract is up, you will not get high speed 3G/4G service unless you are in a "refarmed" area. Technical details aside, T-Mobile plans on doing this to the entire country eventually, so if you're on a 2 year contract with AT&T you MIGHT be okay if you then leave for T-Mobile on the same phone - it still depends on how T-Mobile does the refarming and if it truly will affect each and every tower in every city. (The plan is to have a lot of their major markets done by the end of the year which *checks the date* is coming up very fast)
You can use an unlocked 810 on AT&T in the future with no problem, it uses the same data bands as AT&T does.
You won't get the 822 on either AT&T or T-Mobile... two completely different technologies (GSM vs CDMA). Same with the 810 and the 820 - you can't get them on Verizon.
As for LTE? The 820 has it but it likely won't run on T-Mobile's up and coming LTE... depends on a lot of things within the bands and T-Mobile's request to the FCC to let their LTE bands be interchangeable/useable with AT&T and Verizon. My guess? No.