"dad's there" is also good I think.
I know many people (including myself from time to time) who use "daddy" as the intimate form of father. Couldn't тато also be translated "daddy"?
Daddy is more a cuddly term of endearment, as opposed to dad or father which is more formal in English. I just googled whether or not Ukrainian has an informal word for dad and i couldn't see any results. I guess context would be key in choosing which one to use (if there is a slang term for father) though it's probably best (like any language) to stick with the highest levels of grammar and formality until you get a good grasp on a language.
Why "yes, dad there" isn't a correct sentence? I really have no idea where is verb in this sentence. I'm native polish speaker and this sentence will mean the same in polish so I guessed that i'm right
In English, the "to be" verb is necessary when expressing location (called in linguistics a "copular" verb). I know that in the Slavic languages, the "to be" is often dropped in the present tense, but it is a necessary part of an English sentence. You would need to say "Yes, dad is there" or "Yes, dad's there".