I wrote "jest tutaj" instead of "jest tu" because "tu" is a short version of "tutaj" (there are many words like that in Polish, for example "mi" and "mnie", "się" and "siebie", "ci" and "tobie" etc.). Tutaj is longer, then the accent goes there and you can emphasise the fact that you are HERE (not there). "Jestem tu" - in this sentence the accent goes into the second word but "tu" is very short, so you should use "tutaj". In other case it would sound unnatural for Polish speakers. Of course, it's not very important but if you want to speak Polish perfectly, then remember about it.
The same goes with other words, for instance:
- Kocham cię. = I love you.
- Kocham ciebie, nie ją. = I love you, not her.
- Ciebie kocham, nie ją! = It's you whom I love, not her!
Cię kocham is incorrect.
And yes, tutaj and tu are the only 2 forms of here. There are no more. :)
To make it easier, look at it:
- Jestem tutaj. (I am here.)
- Tutaj jestem, nie tam! (I am here, not there!)
- Tu jestem! (I am here!)
It's not exactly wrong, but it sounds strange to me. Generally, for 'not being somewhere' Polish uses a completely different construction. "Mojego taty tu nie ma"/"Nie ma tu mojego taty". So "my dad" takes Genitive, and there rest of the sentence would be literally translated as "There has not... here". Which sounds absurd from the point of view of English, but that's how it works in Polish.