Hi,
It looks nice, but then they all do! I trust that you got it for a bargain price at this time of year.

I don't know about an '87 but no doubt the experts will be along before long to give more advice. My comments here are based on ownership of a 2001 Yam TT600R which has a somewhat similar engine, hence it is a "thumper" with the same vibration characteristics.

The rubber inserts are to reduce vibration of the engine cooling fins - dampers basically.

To find oil leak sources, first throughly clean the suspect area - the leaked oil usually looks much worse than it is because it spreads so much over any available surfaces. Some people say to spread talcum powder on the engine surfaces and spot where the oil shows; when I tracked down sources of leaking oil, I thought the powder would blow away in the slipstream too easily (it did not leak when static at tickover), so I stuck masking tape (the sort of stuff used while painting) on the engine and this stained where the oil was leaking.

Do check your oil level of course, as it is leaking right now, and I suggest that you have a look through the posts in here about workshop manuals.

Sounds as if it may warrant a servicing; for instance, sort out the cold starting on the button, clean carbs, all as you suggest.
I can't say that I would like to be doing that without somewhere to spread out the bits etc - without a garage, it looks as if it will be your household table top?! You would not be the first to keep your bike in your house!!

On the 3AJ engine there is no cyl head cover gasket; it is sealed with a silicone based goo each time you take it off, which requires the old stuff to be cleaned off before applying the new goo - bit of a pain on the second or third time of doing this!!

Hope this helps a bit,

__________________
Dave