When does the water get warm enough to swim in?

Yes

I would say late June

Definitely when the sun is out, it's fed by run off so it can be quite cold at times but summer heat works well with it

I'm not entirely sure. However you can check the temperature and status of the park at this link. https://stateparks.utah.gov/parks/palisade/current-conditions/ This is updated pretty often so you will be able to plan a trip accordingly.