What happens if I gain control of my opponent's Rekindling Phoenix until end of turn, and it dies?
You get the 0/1 token, but you don't sacrifice it unless you have another Rekindling Phoenix in your own graveyard.
The Phoenix's triggered ability triggers for the current controller of the Phoenix, which is you. Therefore, you put the 0/1 token on the battlefield under your control. The Phoenix, however, goes to its owner's graveyard, which is your opponent.

109.5. The words “you” and “your” on an object refer to the object’s controller, its would-be controller (if a player is attempting to play, cast, or activate it), or its owner (if it has no controller). For a static ability, this is the current controller of the object it’s on. For an activated ability, this is the player who activated the ability. For a triggered ability, this is the controller of the object when the ability triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed triggered ability, see rules 603.7d–f.
404.1. A player’s graveyard is their discard pile. Any object that’s countered, discarded, destroyed, or sacrificed is put on top of its owner’s graveyard, as is any instant or sorcery spell that’s finished resolving. Each player’s graveyard starts out empty.

While the Phoenix lands in your opponent's graveyard, the ability triggers for you, because you were its last controller before it left the battlefield:

603.10a Some zone-change triggers look back in time. These are leaves-the-battlefield abilities, abilities that trigger when a card leaves a graveyard, and abilities that trigger when an object that all players can see is put into a hand or library.

You own and control the 0/1 token, because (in this case, not generally) you controlled the effect that created it:

110.5a The player who creates a token is its owner. The token enters the battlefield under that player’s control.

Since the triggered ability of the token requires a Rekindling Phoenix card as a target in your graveyard, if you don't have one in your graveyard, the ability immediately leaves the stack without resolving. It will keep triggering that way on each of your upkeeps. It will not find the Phoenix that has landed in your opponent's graveyard, and the effect will not return the Phoenix.
Note that if you had a Rekindling Phoenix in your own graveyard, for example because you had to discard it, or the opponent destroyed your 0/1 token before you could sacrifice it, the token generated by the opponent's Phoenix would allow you to return your own Phoenix. That's because the token's ability is not directly connected to the Phoenix that created it, and allows you to find any Rekindling Phoenix in your graveyard, no matter how it got there.
The same goes for your opponent. If he took control of your 0/1 token and then later sacrificed it to its ability, he would get his Phoenix back to the battlefield, whether or not the token was generated by a Rekindling Phoenix he owns.