To your first question:
If Phoenix is negated and destroyed when special summoned or normal summoned from your hand/deck/RFP then his effect will activate and he will be special summoned in your next Standby Phase.
Now this has to do with your second question too.
However, if Phoenix is negated in the grave, it is NOT destroyed since it was never special summoned, and therefore it never left the grave. And for this reason, if Phoenix is negated when special summoning himself from the grave then he won't come back.
If he is special summoned on the field and THEN he's destroyed (Torrential Tribute for example), then he will be special summoned in your next standby phase.
Also note that you cannot negate special summons through card effects with Solemn Judgment or Horn of Heaven, therefore when Phoenix is summoned through his own effect from the grave he can't be negated by a Solemn Judgment or a Horn of Heaven.
Locking this upon confirmation that you understood.