Just adding my 2 cents as we LOVE visiting Jackson and the Tetons!
As for the Old Faithful Inn...you won't miss the TV's! Really! It's a different pace out there, and there are sooooooo many other activities to do instead of sitting in front of the tube. Even the kids. It's GREAT. No AC is not a problem, unless you're there in the middle of summer during a 'heat wave' (we've had that happen - can get a bit toasty but they usually have ceiling fans that help).
As for May...I guess it just depends on the type of weather you want (and what you are used to). If you want warmer weather, I wouldn't go in May. I've been in mid-late June and had cold and sleet, I would think that May would be just too iffy in the weather dept. I've been told that if you want dry (as in the ground, not muddy from the winter run off) and warm, go in July or August. I've been once in June, 4 times in July and once in Sept. I prefer July. Of course July is more crowded - but it never bothered us as we just headed out of town into the parks and hit the quiet, low traffic trails.
Anyway, I could type for hours about Jackson.....love love love it out there. Our absolute favorite!!

It's beautiful. Breathtaking. So much to see and do. And I'll ditto tacomaranch on a few things....definitely hydrate, start on the flight out. And definitely do some walking/exercising before heading out there, or you'll be paying if you plan to do some hiking. Jedidiah's is good, but I must say I prefer the Bunnery.

And then there's always a quick bagel and coffee from Pearl Street bagels before heading into the park. And don't miss Bubba's! And catch a show at the Mangy Moose! (See, I can't stop!)
Have an awesome time!!