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.