As far as those residential modifications go, either can do the work. A GC can do much more extensive construction as well as building from scratch.

Home improvement license is what a normal tradesperson would get as you need this license to do any kind of repairs or remodeling. You require a $20,000 bond.

General contractor has more stringent requirements and can actually build or demolish buildings - hence permitting through the Building Department versus Consumer Affairs.

I have no idea what your building's CCR's or lease if it is a coop requires but you should check with the management in terms of what is required.

In my building, you need to file a construction plan and sign an agreement. All contractors need to be licensed and have insurance of $1,000,000. I would be surprised if a condo or coop in Manhattan didn't have equivalent construction restrictions. So if you don't know what those are, that should be your first first line of inquiry,