If I'm remembering the sequence of events correctly, Rey lands Ren's TIE on Ahch-To and burns it, intending to go into self-exile. Luke's Force ghost shows up and convinces her not to, so she pulls the Sith Wayfinder out of the wreckage of the TIE fighter (it's apparently immune to fire) and Luke dredges up his old X-Wing; Rey takes the X-Wing and the Sith Wayfinder and flies to Exegol.
So, your questions in order:
- In Luke's old (apparently waterproof) X-Wing.
- It's apparently fireproof.
- The Wayfinder was in the TIE fighter when she stole it from Ren, and she hadn't removed it.
Note that there appears to be some confusion about Ren's TIE fighter; he loses one on Pasaana, but he arrives on Kef Bir in another one. This last one must have the Sith Wayfinder, since he tells Rey "The only way you’re getting to Exegol is with me." And it's quite clear in the movie that she removes the distinctive pyramidal shape of a Sith Wayfinder from the smoking wreckage of the TIE fighter just before Luke pulls his X-Wing from the ocean.