I had this exact problem when I changed jobs and then ultimately FIREd in 2019.
I moved everything into Fidelity which has a zero-fee HSA. Zero transfer fees, zero administrative fees, zero requirement to carry some dopey cash balance. It is an ideal solution, found through this forum (love that MMM collective knowledge!). There are even some Fidelity zero-expense funds you can invest in but I went for the VTSAX equivalent.
I plowed it all into FSKAX, made my 2020 full contribution as the markets sank in 2020, and lived happily ever after.
I am not sure if this handles your Question #2 since I have an HDHP post-FIRE so I can continue to invest that HSA contribution. The Collective can chime in.