We are planning to visit in May of next year. Are there trails that are open? Is it still snowy at that time of the year?

It will likely still be very snowy, however many of the lower elevation trails will be open. Expect to hike through snow and mud. The Going to the Sun road will not be open, you will have to plan extra time to travel from east to west.

There will be trails open, but the going to the sun road will not be fully open generally at that time. It depends on how much snowfall has come through the winter.