There is no Cat/class limitation on Ground Instructor Certificates. As for your other questions, in 14 CFR Part 61.213(b)(1) we read:
(b) The knowledge test specified in paragraph (a)(3) of this section is not required if the applicant:
(1) Holds a ground instructor certificate or flight instructor certificate issued under this part
It doesn't specify that the CFI ticket need be unexpired.
in 14 CFR 61.199 (reinstatement of an expired instructor certificate) under paragraph (b)(2) we read:
The holder of a flight instructor rating or a limited flight instructor rating on a pilot certificate may be issued a flight instructor certificate with the current ratings, but only if the person passes the required knowledge and practical test prescribed in this subpart for the issuance of the current flight instructor certificate and rating.
So, it appears, that yes, once you take or -II tests, your I will be reinstated as well.