Answers:
That is your siphon assembly, possible it might need replacing
I would try to tape it once if i was going to use tape at all. as having decorated for over 20 years i would always suggest to use tape as a last resort
Sillicone can be used as a last resort but i would always suggest putting back what the original was as the seal was designed specifically for that purpose … unfortunately to renew the proper seal you will have to disconnect the plumbing and waste and take the sink fully out the reseal it properly
Could be the mechanism on the hinge has started to fail i would ask is it a door seal or a window seal if it’s door it can be adjusted if it’s a window it might need replacing