They are different things. You use есть if possession is the point and omit if it isn't. Есть is not typically used with medical conditions, body parts and when the sentence can essentially be restructured into the one that has no "have" :
- У неё температура. = She has a fever.
- У меня голубые глаза. = I have blue eyes.
- У меня большой стол. = I have a big desk = My desk is big (if this is what the original sentence meant)
There are a few other scenarios where the identity of the thing possessed is more relevant than the fact of possession. It is just that the translation is the same: Russian makes this distinction in the present tense and English does not (in real life you know what the message of your sentence is, so you select the structure that makes sense).
In the past and in the future Russian uses был(была, было, были) and будет (будут) regardless of what you mean. So, there is no choice there.
У is a preposition meanig "near, at". It isn't used in the spatial meaning with living beings, though.
Russian technically does have verbs that mean "to possess, to own". However, the most common way to express having in the neutral/spoken language is to use the verb "to be". Then you say that an object is "near" you (lit."By me there is a cat" → I have a cat). У is a preposition that requires the Genitive case, hence the difference in form:
- I have a banana = У меня есть банан.
- Mom has a purse. = У мамы есть сумочка
- You have a dog. = У тебя есть собака
- The dog has a tail. = У собаки есть хвост.
Иметь and обладать are mostly used in official and formal styles, though иметь also has a few uses in neutral set expressions (e.g., "Я имею в виду, что .... = I mean that . . .").
It is not "possessive" that matters here. У is a preposition, and prepositions know what they want in terms of cases. Such prepositions as у, около, возле, от, из, из-за, из-под always take the Genitive. It does not matter what the meaning is even. For example, около can mean both "near" (literally: in the spatial interpretation) and "about" (approximately):
- около магазина = near a shop
- около 5 км (pronounced "около пяти километров") = about 5 kilometers
A few prepositions will use two cases depending on the meaning. Very rarely there is a niche third use that requires another case (typically so unique that a non-native may learn it some time much later). Of such rare occasions, only в currently springs to my mind.