Do Americans perceive that kind of variety in US accents? I can tell a Texan accent, or a Deep South/New Orleans accent, or a strong New York accent

Yes, we can perceive that variety quite clearly, but only to the large general regions you just specified. Tight regional speciation like the kind you described for Liverpool/Manchester/etc is only something that experts can discern.

My favorite US accent to make fun of is the mid-northern Minnesota/North Dakota accents from communities made up of mostly Norwegian descendants. Don'tcha know. I just love watching "Fargo" and listening to "A Prairie Home Companion".