When would I use this belt, especially if I play solitaire?
Most likely during the Planning Phase. Otherwise, if you have an Event card you want to play at some point but need resources for it from a sphere you don't otherwise have, you can exhaust the belt then to gain the resources you need.
If I exhaust the belt during the Resource Phase, what good is it?
Probably none, unless you have an Event card you want to play during that phase.
If it's exhausted during the Planning Phase, how do I use those resources for the chosen icon if I've already played allies and attachment cards?
You can take player actions at any point in the Planning Phase, so you can exhaust the belt to give your character the sphere icon you need, then play cards using that character's resources for that sphere.
When playing solitaire, does 'First player' and 'Next player' all apply to me?
In solo, you are both the first and last player.