Do washer hookups need that rubber gasket/washer?

I recently got a new washer and the installers installed it on the wrong cold water intake (the non-working one.) When I transferred it to the “right” one I found that it would not screw in unless I removed the rubber ring inside the hose, after I did it fit perfectly.

I have not turned the cold water on, though. When I do so should I expect a massive leak?

yes. required.

Agreed, although you might be able to get away with just using teflon tape on the threads. Doubtful.

yea your gonna need it, also when you first try and put it on turn it 2 the left till it drops on a thread then tighten it