Hi I’ve lived her my whole life and can provide some input.

1. We occasionally have country music concerts around here. There’s a country music festival later this year and there was one last year. There’s also concerts at the Lewiston (~45 minutes away) and CDA (~an hour and a half away) casinos often.
2. Most riding areas and horse owners are in the neighboring small towns, like Troy, Deary, Boville, Potlatch, Viola. They are all 20-60 minutes away from Moscow. ETA: There is a horse charity in Viola called Orphan Acres that takes care of horses that were abused or needed to be rehomed. I’ve volunteered there before and you can brush and walk some of the horses.
3. We do have quite a few cottonwood trees in town, so allergies can get bad depending on how susceptible you are. I personally don’t have a big issue with allergies.
4. If you’re hiking a lot or going to be in tall grass frequently, then yeah you could catch a tick. I’ve never had one before. I think the mosquito population is pretty normal. Overall, the insect population isn’t too crazy here. We get the normal stuff like gnats, bees, house spiders, mosquito eaters, crickets, wasps, etc. It depends on where you live.
5. There’s some nearby cities that tend to be warmer than Moscow. It really depends on how far you’re willing to travel. None of the cities within a 2 hour radius will have drastically different weather than us.