One Korean friend says yes, one says no. Based on that, I assume if there was any difference in pronunciation, it's lost to history, or it's like a chinese tone. Considering how often Koreans misspell words with 애 and 에, I'm going to guess the former.

With 게 you widen your mouth more but nobody does it

Your mouth shape should be slightly different. 개 is a wider smile than 게

Some dialects pronounce the vowels distinctly. Some dialects not so. Some dialects treat both the same and infer the word by context.

I think it's like the difference between é and è in French (Google "French phonology")

Should be dragon if my last question was accurate. I see a lot of really short words pass as typos