In English, "coat" and "jacket" are often used interchangeably. The main difference, I would say, is that coats can be longer and more formal. E.g., a man's wool coat that goes down to the knees would never be referred to as a jacket (the same is true for a "trenchcoat"). But coats are not necessarily formal (e.g., "winter coat"). Jackets are, however, almost always waist-length and relatively lightweight.
Bottom line: "coat" is more of a generic term, whereas "jacket" has a narrower range of uses. I.e., articles of clothing that can be referred to as jackets are often referred to as coats, but many items of clothing that are coats can never be referred to as jackets.
Yes, which is why I gave the translation that I did, earlier in this discussion - excluding women apparel, when Poles say "płaszcz" they mostly mean trenchcoat of the type worn by Chandler protagonists. ;-) Generally, if it goes beyond the knee, it is a płaszcz.
As for women clothing, things aren't as clear cut, but I doubt anyone would call a "płaszcz" anything that isn't at least reaching half of thigh. And even then, some might call it a "płaszczyk"(diminutive of "płaszcz", so small coat) instead. ;-)
Generally, it's better to think that you always change the form. Some forms are just identical. And indeed, for plural, most nouns have Accusative = Nominative.
Only masculine personal nouns (denoting a group of people with at least one man) have plural Accusative = plural Genitive.
Some nouns in English (and in Polish as well, by the way) are so called 'pluralia tantum', they do not have a singular form. For example you cannot say "a underpants" or "a pants" or "a trousers". It's "a pair of..." then.
But the majority of nouns are countable, and then you need an article in front of them, so it's "a jacket".