Trouble starting when cold

[Khumarahn]

Hi guys. Sorry for one more topic on starting in cold weather, but I couldn't figure out the answer in my situation.

I have 2003 gz 250, and I have to follow quite a procedure to start it at around 40°F or colder.

First of all, it works fine when warm. When cold, I have to
1) use choke
2) turn fuel valve to "PRI"
3) use 1st gear (doesn't start in neutral)
then wait half a minute, or try to rotate the engine by pushing bike forward or backward; and then it starts fine.

I am total noob in motorcycle repairs, but I believe I have straight hands. From what I understood, battery may be the problem. I tested it in Autozone, and it was bad, and I replaced it with a new one. But the problem persists.

What should I try next? Clean the carburetor? (what cleaner to use?)

There are also traces of oil on the grill, as if some gasket on top of the engine is leaking... I wonder if this is related.