Yes, you can open an HSA with another provider. Is your company making contributions into your HSA? Are you making contributions via payroll deduction? It would be best to limit your contribution to just your company sponsored HSA and then do a rollover into another HSA of your choosing, if you feel it’s worth the bother.
Your spouse can open an HSA and contribute to it only if her health plan is a HDHP. I’m a stay-at-home spouse who is covered under my husband’s family HDHP. I just opened an HSA of my own at Fidelity only to take advantage of the $1,000 catch-up contributions for 55+.