I live in Twin, although I am from back east. I am curious where you are from and why you are looking to move here? Is it for a job, family or....?? Have you spent any time here? It is not for everybody. I would encourage you to really understand the city and the culture (forget about the biking) before relocating. It may be a very good fit and may not be at all.
To answer your questions:
Mountain biking opportunities are decent. Twin is an ag town and there are very few places to ride close to the city. The notable exception is the Auger Falls Park which is very new and very cool. It is in the canyon, only a few minutes from anywhere in town and great trails for a couple hours of riding. Many riders are down there several times a week. There are other threads about it on this forum.
The other place we typically ride is in the South Hills at Indian Springs. I live in the middle of town and it takes about 21 minutes to get to the parking lot. The trails there are much longer and involve a lot more elevation change. It is a good place to ride but ultimately a bit limited. Again, you can find Google maps of the trails and discussions on this forum.
Trails on more of a "destination" level can be found in the Wood River and Sawtooth valleys to the north. Think Bellevue/Hailey area all the way to Stanley area and beyond. The center of this area is Ketchum/Sun Valley and it is 1:45 drive time from Twin. There are numerous opportunities to ride there, from short evening rides right out of town to mega epic backcountry opportunities. There is a lot, a really lot. BCRD Trails can give you some idea.
Boise has a ton of trails right out the backdoor. I don't know a whole lot about riding there. You can get maps. It is 2 hours + and 125 miles to get to Boise from Twin.
As far as season, the good stuff up north is generally open from mid May to around the beginning of November, give or take a bit depending on weather and which trail we're talking about. Some are not accessible until well into June given elevation/snow. This year things are opening up a bit earlier than usual I think. The Indian Springs trails have a longer season, March/April until November. At Auger Falls we were able to ride every week of the year through this last winter. The desert sucks up the winter moisture pretty quickly. Of course this can vary with snow fall and temperatures.
Hope this helps.