This is a special construction to indicate possession. "U menya est' moloko" would literally translate to: To me (U menya) there is (est') milk (moloko). "Est'" in this case means "to be". The verb can also mean "to eat". You are right about the simplification. Remember, in Russian, the verb "to be" is not necessary in the present tense. Hence, the omission is also correct. ;)
I really like this Russian structure to say "I have ...", it sounds kinda like a very important thing to say. In Spanish it's just one puny word: "Tengo" (plus the pronoun before this if you must include it, but we usually don't, since the verb already tells you which pronoun you're using.
Correct answer would be 'у меня есть', but it's very specific. You can use this to emphasize the possession, like 'I do have a thing'. Overwhise the answer should be just 'да' or 'да, есть'.
'меня есть' without the preposition 'у' literally translates as 'to eat me', which is probably not what you're going for.
"Есть" can generally mean two things.
- "Есть" - infinitive of the verb meaning 'to eat'. "Ест" is a masculine present tense form of the verb "есть".
- "Есть" - a form of a verb "быть" ('to be'). This form called 'realis mood', in case you want to make your own research, but unless you're really interested in linguistics I would advice against it. What you need to know is that "есть" might have several meanings depending on the context: to eat, to be, to have. It might look confusing at first, but in practice it's always quite obvious.