Trash pretty much anywhere in the fuel system is going to make problems. The closer it gets to the heart of the carburetor, the more problems it makes. However, it is very unlikely that trash in the location you mention would cause the problem you describe. If it were blocking the flow of fuel, the engine wouldn't run even with the choke closed. I think the carburetor exhibiting that behaviour has some clogged internal passages that need to be cleaned out.
There's nothing wrong with a properly refurbished carburetor. "Remanufactured" carburetors (and other parts) tend to be junk.
Yes, that's the rubber tip of the inlet needle. It's supposed to be there.
Probably not.
