Does a previous AFSP approval get invalidated when making a new training request?
Yes, you can have multiple requests open at the same time. This is from the AFSP Candidate Application Guide (step 7):


  It is possible for a Candidate to have several active training requests at a given time. These requests may be for the same or different flight training providers. Each training request form will be processed separately; AFSP approval is valid only for the Provider listed in the application.