Yes you are allowed to wear it and its not a problem. As a prove you can test it out during the week by leaving it laying around for 24 or 25 hours without winding it and see if it would still work. This would prove that they are capable of working through out the whole shabbos without you winding it and so you aren’t doing anything by wearing them.
I’m sure there are some poskim who will disagree so you have to ask your LOR.
I am pretty sure shmiras shabbas kihilchasa speaks about this,and I think it is mutar only if it is mechanically done through the movement of your hand.
R Shloime miller of toronto paskens that they may be worn on shabbos provided that theres already enough “juice” in the watch that whatever winding it gets on shabbos isnt needed. i believe you also cant shake it on purpose; this psak, i believe, only refers to normal wear.
It depends. Most work on a spring action which winds the spring, these old-fashioned types are Mutar as long as they are running when Shabbos begins. However, the Seiko Kinetic (and maybe others) generate electricity by the movement of your wrist, and these should not be worn on Shabbos (even though their may be grounds for leiency).