"You have a very short surname."
Translation:У тебя очень короткая фамилия.
«Есть» is used when 'having' or 'existence' is a part of the new information. So,
- «У тебя есть очень короткая фамилия» tells use two pieces of new information: 'You have a surname. Your surname is very short.'
- «У тебя очень короткая фамилия» tells use one piece of new information: 'Your surname is very short.'
«У тебя есть очень короткая фамилия» sounds as if people might have several surnames, or might have no surnames at all. While this is true in some cultures (I think Mongolians often go without any surnames), this is still pretty rare, and most Russian speakers are used to the fact that one person has exactly one surname, no more and no less, so this sentence sounds very unnatural in Russian.
Коро́ткая is a native Russian word (коро́ткая is the feminine form; masculine is коро́ткий, neutral is коро́ткое); кра́ткая (masculine кра́ткий, neutral кра́ткое) is the same word borrowed from Old Bulgarian (via Church Slavonic). They mean roughly the same thing, but in some cases коро́ткая sounds more natural, in other cases кра́ткая. (However, if you replace one with another, you will still be understood.)
Ко́роткая is more common in daily speech, while кра́ткая sounds more formal. When talking about length, кра́ткая often has the meaning 'concise, condensed' (therefore кра́ткая фами́лия doesn’t sound too well).