Is Escardó's metric semantic for PCF+timeouts fully abstract ?
Regarding your second question, I seem to remember that for higher-order types the question was linked closely to whether PCF+timeout was equivalent to Type Two Effectivity (Turing machines with infinite inputs and outpus), i.e., Kleene's second partial combinatory algebra. John Longley claimed for a while that Kleene's second algebra was equivalent to PCF+timeout+catch, but in the end he never published a detailed result.

On the other hand, I am pretty sure that John Longley opus magnum "On the ubiquity of certain total type structures" (Mathematical Structures in Computer Science 17(5) (2007), 841--953) implies that the higher-order functionals definable in PCF+timeout are precisely the hereditarily effective ones.