Exhaust fan dripping

All the things you've done should have helped some, except the duct boosters. They can't add much flow to the system unless the return ducts were increased proportionally (or similarly boosted). You get increased flow by increased pressure or larger ducting. Keeping upstairs doors closed will help by slowing the 'fall' of cooler air downstairs. Try leaving fan "ON" for a 24 hr period to see if your house will equalize better.

You can force the system to concentrate on the upstairs by reducing both the supply and return vents in the main floor and basement. You may have to do some makeshift blocking (reduction ) of the returns with plastic sheeting and 'no residue' duct tape. If successful, I believe magnetic covers are available.

AFA blocking above the upstairs ducts, by all means. Save yourself some expense (and trouble, that much foam is hard to scaffold in a vertical shaft) and make 2x4 blocking (essentially fire blocking) to go above the vents. If you have typical framing 16 inches on center, a 14 1/2 inch piece will span stud-to-stud. Use a pair of pocket screws on each end to secure the blocking in place. Use fire caulk/foam for any gaps.

Probably the best solution in your case it to directly vent out. Since you have two external walls one of the walls will allow you to vent out of it. You need to pick the one that runs perpendicular to your joists.

There is not much downside in venting out. It is easier. By venting up you are actually causing an opening in your house for hot air to escape - hot air rises, will push open vent cover, and release. With a side vent you don't have those issues. The only issue is directing the moisture away from your house but this depends on the type of exterior you have.

You will pick a place for your fan - by the shower, in-between two joists. You will point fan vent towards wall that you can vent out of. You will run duct from fan to wall. Cut hole in wall slight bigger than duct (usually 4-6" in the US). You will cap it from the outside with a vent and secure vent to outside wall. Spray foam the openings/cracks created. That part of your fan install is done. Still need to get electric up to it.

To cover your other points. You can vent out the basement but you should not have a fan anywhere other than a ceiling. Anything higher than your fan will get condensation and could eventually mold.

Edit: Forgot to mention. Give your duct work a tiny tiny slope down so any moisture will drip out instead of back to your bathroom.