Yes, absolutely. Simply check. They’ve spent far over a billion over the past decade directly to politicians, PACs, etc. The teachers union is the largest union in the US, and I think even the world.

Simply google the question, plenty of sources.

Yes, they are politically active, and that includes donations of money and in kind.

Yes. In my state’s last cycle, the state teachers’ union was the largest direct donor to state-level candidates of the winning party.

https://www.opensecrets.org/industries/indus?ind=L1300

https://www.opensecrets.org/industries/indus?ind=Q11

Yes. And the use of money and undue influence of money are the problem-- not the idea of lobbying per se.