1. When I upgrade a unit, can I add any extra items or traits? For example, if I have a custom unit with a spear and I upgrade him, can I add a horse?
Not if by unit you mean one created in a town, etc. You can do this to a champion or your sovereign, of course.
2. If I neglected to click the upgrade slot icon when designing a unit, can I still upgrade that slot? Or am I stuck with whatever equipment the unit started out with?
3. Can I switch weapon types when I upgrade? For example, if I start off a unit with a spear, but then decide the unit actually wants an axe, can I upgrade him from spear to axe? Or am I stuck with a spear?
Once again, you're stuck.
I'm hoping more flexibility will be added to this game element in the future, but we'll just have to wait, and see.