How to implement invitations limits per user when using the Invite Module?
Invitation expiration

Quote from the Community documentation about the Invite module (from its installation instructions):


  Invitation expiry: Set the expiry period for user invitations, in days (default select up to 365 days).


So the answer to your 2nd question do unclaimed sent invites expire? is YES. That's what's stated also in comment #3 of the issue about Set expiry date to never.

Setting invitation restrictions

Looks like this feature "used to be available in a prior version of the module". And according to comment #3 of the issue about Limit invitation per user/role, it should be back some day.

While waiting for that to happen, you may want to use some type of Rules integration. Such as the solution as included in the issue about Rules integration for D7. It includes a patch for the 7.x-2.x version (status "RTBC"). If you're using 7.x-4.x however, then note the issue status "Needs work" ...