Seasonal gear used on non seasonal characters?

Once the season is over, yes.

While the season is active, think of it like having 2 separate Diablo 3 accounts: one for Seasons, and one for non-Seasons. They are completely separate. (Your hardcore and non-hardcore characters are also separate.)

When the season is over, all of your seasonal characters are converted into non-seasonal characters. The first time you log in after a season is over, all of the gear that was in your seasonal stash will be mailed to you. At that point the items are no different from an item you found on a non-seasonal character.

Blizzard always announces the end of a season at least 2 weeks in advance. When they do, they write a new blog post explaining exactly what happens when the season ends. For example, here is the blog post from the end of last season: Season 10 Ending Soon.