у меня есть = I have (smth) у меня = the same thing, but without "have". It means something that belongs to you. You can say «у меня всё хорошо» = «everything is ok» or «у меня сегодня день рождения» = «it's my birthday today»

Part I. Gabriele, есть emphasizes presense of something as opposed to absense. У меня есть борода. I have a beard. (you don't). У меня длинные волосы и карие глаза. I have long hair and brown eyes (can you say "I don't have" if your hair is short and eyes are green? I think you rather say "mine are green".) If I _required_ to have long hair and brown eyes for some job, I can say both: У меня длинные волосы (more like "my hair is long") У меня есть длинные волосы (stressed "ЕСТЬ", "I DO have long hair"): I do have what's needed. It sounds funny though, as if long hair is something like a book which you might own or not:) And you can say "I don't". When telling "I have brown eyes" I'm answering an (unasked) question: Which colour are my eyes: green or brown? Not if I have a brown pair in addition to a green one. This is why no "есть".

Well, there are many nuances. I'll try to give some examples. 1) У меня есть собака - I have a dog. But: Я не могу уехать надолго — у меня собака = I can't leave for long - I have a dog. It's not just about an ownership, it's the reason why I can't leave for long is my dog. 2) У меня есть парень = I have a boyfriend. У меня новый парень = I have a new boyfriend. Very often, when there is an adjective, we don't say "есть". Though I think that sometimes it's possible. Here is the example: У моей сестры есть образование. = My sister has an education = she's educated. Here we can't say У моей сестры образование. But: У моего брата высшее образование = У моего брата есть высшее образование = My brother has a higher education. I think that the difference here is very subtle and it would depend on the context... The first one is quite neutral, and the second one is more contrastive: У моего брата есть высшее образование, а у меня нет. Or: Вообще-то у моего брата есть высшее образование = In fact, my brother has a higher education. There are some contexts where we can't use "есть" at all. Here they are: У меня зеленые глаза = My eyes are green. У меня хорошее настроение = I'm in a good mood. У меня температура = I have a fever. Не могу сейчас говорить, у меня экзамен = I can't talk to you right now, I have an exam. Right now I'm sitting in a classroom and take the exam. Hope that helps! This question is really tricky, I advise to remember some key examples that are important for you and use them :) There is a good song to memorise this construction — "Место для шага вперед". It's the song of the iconic band Кино.

Это некорректное определение. Ничто не мешает сказать: «у меня кот» вместо «у меня есть кот». Или: у нас газовая плита. И все это в смысле владения.

You can use ‘У меня есть...’ to talk about things and ideas you have: У меня есть собака. I have a dog. У меня есть идея. Пойдем в кино? I have an idea. Let’s go to the cinema! The phrase ‘У меня...’ can be used to talk about events you’re participating in, conditions you might have (esp for illnesses) У меня грипп. I have a flu. У меня болит голова. I have a headache. У меня сейчас математика. I am having a Math class now. У меня завтра день рождения. Tomorrow is my birthday. So, ‘У меня есть’ is about possession, ‘У меня...’ is about sth happening (or about to happen) to you. Hope that helps.