Yes it is possible and will depend on what is specified in the operating agreement.

2 member LLC in California providing occasional services. It's very part-time. Do we get 1099s AND a K1?

Only K1s need be issued. No need for a 1099.

Have the board appoint you as an officer.

May I use the registered trademark sign (®) in my company's logo?

Yes as long from the placement it does not suggest that the logo portion is a registered trademark.

Is "smartphone" and "tablet" specific enough in a patent?

You will need to claim the structural components of the phone in terms of hardware and/or software that gives rise to the benefit of your inventions.

Does my business name that I'd like to trademark, use common or descriptive words that would be grounds for a USPTO denial?

Yes because the marks are descriptive and lack distinctiveness. You can register on the supplement register and if you use the mark continuously...

It depends on the terms of the trust.