There're is common in speech, at least in certain dialects, but you'll rarely see it written. If I were being pedantic, I'd advise you to use there are in your example, because there is is definitely wrong, so there's could be considered wrong as well. But a huge number of English speakers, even those that are well-educated, use there's universally, regardless of the number of the noun in question, so you will probably not receive any odd looks for saying or writing there's, and if you do, just cite the fact that it can't be incorrect if a majority of people use it. As for me (a native New Englander), I use both, but may use there's in place of there're if I'm speaking quickly.
Working on the principle that "correct language" is defined as the common usage among native speakers of the language: "I had a fever" is correct. I don't think I've ever seen "I had fever" in print and very rarely spoken.
In general, I think the rule is that when you are referring to something that is countable, you use an article. When you are referring to something that is not countable, you do not. Thus you would say, "I had a dog" or "I had the chair", because we can count dogs and chairs, i.e. there might be one dog, two dogs, etc. But you say, "I had food" or "I had happiness" because you cannot count "food" or "happiness", i.e. you would not normally say "two foods" or "two happinesses". Think of "a" and "the" as taking the place of the number "one".
Fever is countable. You could say "I had two fevers this year: one in January and another in March."
Which brings to my mind an interesting point: Some diseases are not considered countable. You wouldn't say, "I had two leprosies" or "I had two diabeteses". So we don't use "a" with those. I'm not quite sure what the rule is. Maybe: Long term chronic diseases are not countable, but short term diseases are? Like, "I had a fever", "I had a headache", "I had a cold", but "I had leprosy", "I had cancer", "I had manic-depressive disorder". Hmm, this is starting to sound like, "I have hypochondria".
Best Answer
He asked how I was is the more usual idiomatic form. He asked how I am is a possibility, but it suggests a more continuous sense of "I am".
He asked how am I is not grammatical, punctuated like that. He asked, "How am I?" is a soliloquy, in which he poses the question to himself.