How does promotion work for Military Technician (GS Civilian Job) that requires USAR membership? Do you get to go to BLC?

You have to be an active drilling member to hold a MILTECH position. Yes, you would require a new contract with the USAR. There is no specified contract length, but you have to maintain membership while employed. So if you only do 2 years USAR, you could only work as a MILTECH for 2 years.
USAR membership is a separate commitment from being a MILTECH. Your Reserve unit will be responsible for sending you to schools like BLC and assisting with promotion packets.
There is no set time before you can apply for other positions, but most federal jobs require at least a year experience.

You will need to sign a contract for the army reserves. You will need to be a member of the army reserves for as long as you want to be a miltech. BLC and promotion will be part of the reserve side and the unit your in and not the miltech position. As far as being hired to a different federal agency there is no time restriction, you can apply for and get hired anywhere at any time.

Once you complete your probationary period, I believe it's 2 years now, you can apply for other positions.
While serving as a MilTech, you receive 120 hrs for military leave each calender year, meaning while you're on AD orders for school or AT, you can also receive your regular civilian pay.
Your command will support your military career progression so BLC, ALC, SLC, and MLC won't be an issue. You'll also have a front seat for any available school seats that pop up!