No. If he can't be President, he can't be Vice President.
Exactly. Now, if he were to be appointed VP in the 3rd year because the current VP had died, then I think a former 2 term President could serve as a VP.
No. The limit is not ten years of service, it is being elected President twice or be being elected once and serving more than two years of a term someone else was elected to that is forbidden. Hence anyone who is elected President twice without having already served some part of a term is ineligible to be Vice President, even for a portion of a term that was would not have rendered him ineligible to be elected twice later.
Conversely, it is possible to be President for more than ten years under the following Scenario.
2016: President Alpha and Vice President Omega are elected.
2017: President Alpha dies one month into office and we get three years eleven months of President Omega.
2020: President Beta and Vice President Omega are elected.
2021: President Beta dies one month into office and we get three years eleven months of President Omega.
Despite having now served seven years ten months, Omega has still not been elected President for a term of his own and thus is eligible under the 22nd Amendment to run for either President or Vice President in 2024. If Omega ran for President and won then he would sere a total of eleven years ten months by the end of his term. (Why anyone would choose to have Omega as their running mate at this time is unclear, as if I were President Gamma, I'd be worried about dying a month into office if I had Omega as my Vice President.)
