Question for the Brits:
I’ve heard “mank” and “manky” used as a term for disgusting things. I think I’ve also heard “Manc” used as a term for someone from Manchester. Have I mis-heard? Are they pronounced differently? How do you tell one from the other? Is one derived from the other? Doesn’t that suck for people from Manchester?
‘Mank’ would be very rare and only as some sort of contraction of ‘manky’ which is reasonably common. I suppose if you had a bit of fruit that had gone off you might hear describe the rotten parts as ‘mank’
‘Manc’ comes from Mancunian (or at least it would seem so), I’m not aware of any connection between the two words and I don’t think any connection would be made despite the simlairities between ‘Manc’ and ‘manky’.
To answer questions in the OP:
Yes, a bit on the grimy side, sweaty, unwashed, low flat nasal vowels, incest and violence.
No, more or less the same pronunciation.
Pronunciation wise, there’s very little difference.
In comparison to people from Liverpool, yes.
Same way you tell the other thousand words in the English language which sound the same as each other. From context; it’s easy, honest.
Red - Read
Bore - Boar
Could be worse. There are words both spelled the same AND pronounced the same, but mean different things.
Skate - a fish, or a manner of moving on ice.
Bow - a type of knot, or a gesture of respect.
Row - an argument, or a way to propel a boat.
There’s honestly no difficulty in it. 
Manky is a fairly modern word; the earliest citation in the OED is from 1939 – and that’s in Finnegans Wake and, as with so much else in that work, the sense in which Joyce intends the word is uncertain. The earliest certain citation in the sense of “inferior, defective, dirty, disgusting” is from 1958.
Despite the fact that the word is so recent, the OED thinks its derivation is uncertain. It could be:
from mank, a much earlier Scottish word meaning “maimed, mutilated”, whose own derivation it itself uncertain
a variant of mangy
from mankie a Scottish word for a particular type of woollen fabric which was, apparently, of low grade.
from the French manqué, which has the sense of “spoiled, defective, unsatisfactory”.
There doesn’t seem to be any connection with the abbreviation for Mancunian, though Manc in that sense is of similar vintage, first recorded in 1951.
I think my perception is colored by Ricky Gervais using “Manc” somewhat derisively when talking about Karl Pilkington.
Bow and row both have different pronunciations for the respective meanings you’ve given.
What accents pronounce Bow & Row the same way for both definitions?
Yup. You could say ‘bow = front part of a ship or a gesture of respect,’ and ‘row = horizontal line of objects or a way to propel a boat.’
They’re called homonyms. Manc and Mank are homophones - pronounced exactly the same way but with completely different meanings. It is just context which indicates the difference.
‘Scouse’ has two meanings - ‘from Liverpool’ or ‘a type of stew.’
Worth noting that manky is unambiguous. It’s only mank which can be mistaken for Manc, a person from Manchester. And, as other posters have noted, Mank is extremely rare. I’m familiar with manky, but I had never heard or read mank until this thread. And I don’t think Manc is terribly common either. So I doubt that confusion between the two would arise very often.
Manc’s pretty common in and around Manchester itself.
OK, OK,
Bow - part of a ship or a gesture of respect.
From Gervais I also learned of the word “twonk”, which continues to please me to this day.