У него температура means He has fever. У неё кашель means She has cough. If you ask about saying with есть or without it, just read a dialogue. Доктор: Что у вас? (Literally, WHAT do you have?) Пациент: У меня температура и кашель. Пациент: Доктор, я простыл. Doctor, I've caught cold. Доктор: У Bас есть температура, кашель? Do you HAVE temperature or cough? Пациент: Температура есть, а кашля нет.

When you drop 'есть', it would be a short form but it is still implied. It is only used when it is obvious this is what you're asking. Sorry that I could not come up with a more thorough explanation.

as i understand, you asked the difference between У НЕГО и У НЕГО ЕСТЬ.... formally you need to say У НЕГО ЕСТЬ , but in living speech you can miss ЕСТЬ. regarding to past time you need to say У НЕГО БЫЛА ( there is no way to point past time , instead to use БЫТЬ in p.t) about future time У НЕГО БУДЕТ температура ( the same , there is no another way to point future time instead using БЫТЬ in f.t) But in present time you can miss БЫТЬ ( in present time it transforms to ЕСТЬ)

Все очень просто: когда вы говорите о мужчине, вы используете ЕГО, У НЕГО. Это местоимения МУЖСКОГО РОДА. Когда вы говорите о женщине, вы используете местоимения ЕЁ, У НЕЁ. Например, У НЕГО БЫЛ КАШЕЛЬ. У НЕГО БЫЛА ТЕМПЕРАТУРА. У НЕЕ БЫЛ КАШЕЛЬ. У НЕЕ БЫЛА ТЕМПЕРАТУРА. (Род дополнения не имеет значения)

You can say У него есть температура or У него температура th efirst one is rather a confirmation, that the person has fever. The second one - just a statement - he has fewer. Another example with weapons: У него есть пистолет and У него пистолет the first one might refer to the fact that someone has a gun on him right now, but it just as easily may be a statement that the guy in question is a gun owner, who keeps a pistol in his desk drawer. the second one, however, has more of a warning to it. It states that the guy in question bears a gun right now.