With HMAC, can an attacker recover the key, given many known plaintext/tag pairs? Given many pairs of $(m, t)$, can the attacker compute the key $k$ satisfying $\text{HMAC}(k,m) = t$?  (Assume that $k$ was chosen at random.)
No.  Assuming HMAC is secure against known-plaintext attacks (which is a plausible assumption), the key cannot be recovered.  This follows directly from the definition of a known-plaintext attack; if you could recover the key, you would have a successful known-plaintext attack on HMAC.