When Brynhir Thundersong gets played by Bone Skewer, and is played to a battlefield with Teemo, Strategist, does Brynhir Thundersong resolve first? Or does it not resolve and Teemo, Strategist's ability goes above Brynhir Thundersong in the chain, thus still letting you play cards?

No — Teemo, Strategist's "When I defend" trigger can't get above Brynhir's "When you play me" trigger, so you're locked out of playing cards for the rest of the turn.

Brynhir is played during Bone Skewer's resolution, so her trigger is already pending when Bone Skewer finishes — you go straight back to finalizing pending items before anything else happens (CR 340.3). Combat, and therefore Teemo's defend trigger, can only open in a Cleanup once the chain is empty and the state is neutral/open (CR 344), which is impossible while her trigger is still sitting on the chain. So her trigger resolves first, applies the lockout, and only then does the Combat Showdown begin and Teemo get to trigger — resolving above nothing.

Your one window is while her trigger is on the chain: you can respond with a Reaction card there and it will resolve above her trigger, exactly as with a normally-played Brynhir. After that, "opponents can't play cards" blocks Reactions too, so you can't act during the combat either. Worth remembering that Bone Skewer's play is optional ("you may choose a unit from it") — and she's Stunned, so Teemo's damage can kill her, but killing her later doesn't undo a trigger that already resolved.