Samwell and Jon have a similar conversation earlier (s04e09 "The Watchers on the Wall") regarding Ygritte and Gilly. The oath says take no wife. It does not say have no sex. Jon ended the discussion, noting that the Night's Watch Master-At-Arms Alliser Thorne at the time is not the type to be swayed by loop holes and interpretations.
JON: But if [Gilly] had [offered Sam sex], you would've? You'd have broken your vows?
SAM: The interesting thing is, our vows never specifically forbid intimate relations with women.
JON: What?
SAM: "I shall take no wife" yes, that's in there, there's no denying that. "I shall father no children" it's very specific. But what our vows have to say about other... activities... is open to interpretation.
JON: I don't think Ser Alliser cares much for interpretation.
SAM: Anyway, there's nothing for him to interpret. [Gilly and Sam] didn't.
Plenty of people in the shows have varying degrees of honor bound to oaths. Brienne is absolute. Jamie is less so. The Boltons have none.
Tyrion or Sam for example would be swayed by the "I died so my oath ended" argument. Someone like Jon, Ned, Stannis probably won't.
And you have to take into account that not everyone will believe Jon died in the first place. They don't believe White Walkers exist, when we and some characters do. So even if, legally, the oath is done, some characters won't believe or know it, so they will continue to hold Jon to it. Until someone kills them, as per usual.
As a comparison, Jon was dead for what, a day? Consider it a clinical death, and subsequent resurrection. People are clinically dead, and revived all the time. They are never considered dead for marriage/contracts/bills/property purposes. Not like Jon was declared dead after months/years gone.
That said, Catlin Stark (née Tully) had sworn an oath to the Godswood, to treat Jon like a son, and broke it. She blames breaking the oath to be the cause of the Stark downfall. Swearing an oath to the Gods and failing to keep it has dangerous consequences, and we know magic does exist, so Jon will tread carefully.
As for the last part, it's an entire statement. You can't separate "All nights to come", from the "I pledge my life". And as a whole, the "until my death". As read, if we accept that Jon died, his watch has ended. If we or others don't count his death as a real death (more like a coma, or asleep, or revived), then it hasn't ended, because his life hasn't ended.
Contractually speaking, the death terminates the contract. The "all nights to come" doesn't override it, as it explicitly terms the end of the contract was "death". Now, if Jon was biologically immortal, never dying for 1000 years, then he'd be bound to the oath the entire time. But if he was magically immortal, as in he can die, but comes back to life, again, it's up to interpretation of the contract term of "death" and "my life", as immortality and resurrection are outside the common/plain meaning terms of the contract. When the contract was made, resurrection and immortality were both absurd ideas.
Spoiler Update regarding episode s06e03 "Oathbreaker":
The newly revived Jon states outright in the end of the episode:
Episode s06e04 shows there is confusion and misunderstanding regarding Jon's status:
Jon's second in command who he named the new Lord Commander, Edd, argues with Jon that he is still apart of the Night's Watch. Jon disagrees. Lower ranking members still see Jon as Lord Commander, as the person in charge. Jon strongly disagrees. No one is forcing the issue yet.
