Consider a popsicle, which is basically just water, colour, and flavour: that's also "ice cream" for me in English even though it has no cream.
So basically, if the context is "food" or "snack", it's going to be something like ice cream or popsicle. If the context is "something to put in your drink to cool it down", it's going to be plain ice. If you hear "Eiswürfel", it'll be an ice cube and not an ice cream cube. If you hear "Erdbeereis", it's probably strawberry ice cream and not strawberry-flavoured frozen water. If there's "Eis" on the roads, then you probably just have frozen water, not an accident where an ice cream truck/lorry lost its load. And so on.
If you want to be precise, you can use "Eiskrem" or "Speiseeis" for ice cream, and "Wassereis" or "gefrorenes Wasser" for "ice".
True, but it's more of an "official" type word - you might see it in laws and regulations or on a carton of ice cream, but it's not the word I would use when I say I would like an ice cream or when I offer mine to a friend.
It's kind of like "domestic cat" - it might be the title of a Wikipedia article but I wouldn't tell my friends "I have a domestic cat at home"; I'd just call it a "cat".
You could, and they would; you'd just sound odd.
And I assume you're here to learn reasonably natural-sounding German, not something that people might understand but isn't what they would normally say.
You could say "I am eating a frozen dairy treat" in English and people would understand you. It's not something I'd expect anyone to say or learn in a course, though.
Some verbs change the vowel in the du and er, sie, es forms -- essen happens to be one of them.
This is simply something that has to be learned; you can't guess from the shape of the verb.
(For example, gehen has du gehst but sehen has du siehst; hassen has du hasst but lassen has du lässt; kaufen has du kaufst but laufen has du läufst; pressen has du presst but essen has du isst.)
The only vowel changes I can think of are a/ä; au/äu; e/ie; e/i, but there might be others.