Hello! I'm from the United States, and I can try to answer some of your questions. Present tense for colds: "I have a cold" is most common in my experience. "I caught a cold" and "I've got a cold" are also used. Present tense for fevers: "I have a fever" is most common in my experience. "I'm burning up" and "I've got a fever" also work. If you want to say that you had a cold last night, you can say "I got a cold last night" or "I caught a cold last night". However, the latter two mean that you still have the cold (i.e. you have not yet recovered from the cold that started last night), while the first one ("I had a cold last night") means that you have recovered. If you take away "last night," it still means you have recovered; the listener just doesn't know when you had the cold (i.e. it could have been last night, or last week, or last month... any time in the past). I hope that helps! Let me know if I can clarify anything in my explanation.

1. Present tense: If you are talking about something which you have now, there are two ways of saying this: I have a cold. I've got a cold. There is no difference in meaning between these two sentences. The 'got' form is more informal, but both are fine to use in conversation. Grammatically, saying 'I have a cold' or 'I've got a cold' is no different from saying 'I have a problem', 'I've got a problem, 'I have a sandwich' or 'I've got a sandwich'. These are simply two variants of the present simple form of the verb 'have'. You should avoid using 'got' on its own as an alternative to 'have'. For example, you should not say 'I got a problem. Can you help me?' or 'I don't feel well. I got a cold.' It is extremely informal, and generally considered to be grammatically incorrect. This form is associated with less educated people and with very small children. Do not use it. 'I catch a cold' is not an alternative to the above sentences. It does not tell us anything about how you feel now. 'Catch' refers to the moment when the virus enters your body. The only time you can say 'I catch a cold', in the present simple, is if you're talking about something which regularly happens to you, for example 'I catch a cold every time I go swimming' or 'I catch a cold at least once every winter.' 2. Past Tense: The past tense of 'I have a cold' is 'I had a cold'. The other phrases you suggest are wrong. Even if you don't say 'last night' in your sentence, you are still referring to past time. When we use a past tense, there is always an implicit past time which has been referred to at some earlier point in the conversation or text. And yes, if you are talking about past time and say 'I had a cold', you are suggesting that you don't have a cold any more. All of the above also applies to 'fever'. The idiom 'I'm burning up' isn't the same as saying 'I have a fever'. 'Fever' is a condition/diagnosis, 'burning up' is an informal way of describing a symptom of a fever.

To answer the first part of your question: In present tense, it is more common to say "I have a cold" or "I've got a cold". "I got a cold" is past tense, and "I catch a cold" is never used unless it is a clause (ex. I catch a cold every time I go to Mongolia). For fever, it is more common to say, "I have a fever" or "I've got a fever". Just like the cold, saying "I got a fever" is past tense. The phrase "I'm burning" usually only refers to being very hot (assuming you're not physically burning), but in some colloquial contexts, it is used. Personally, I hear "I'm burning up" more often than "I'm burning". As for the second part of your question: If you want to say that that you had a cold last night, you can say "I got a cold last night" or "I caught a cold last night". Saying "I've got a cold last night" does not really make sense, because the phrase "I've got" in English is equivalent to presently having (i.e. "I've got" = "I have"). If you say "I had a cold" without adding "last night", the general understanding would be that you no longer have a cold. If you said "I got a cold" or "I caught a cold" without adding "last night", there is no suggestion that you do or do not still have a cold, because the person you're speaking with doesn't know when you got/caught the cold (it could have been earlier today, yesterday, last month, et cetera), so the understanding would be based off of other context in the situation.