Only in the first position. If we agree that t' marks a soft T, and that plain t marks the sound it makes, e. g. in French, then the (approximate, but good for practical purposes) transcription will be [t'ítka].
I'd use "цьоця" to refer to the aunt from my Mother's side of the family, And "тета" when referring to the aunt on my Dad's side of the family.
Ukrainian is very specific when it comes to relationships.
Both цьоця and тета are dialectal. (‹Цьоця› is clearly a borring from Polish ‹ciocia›, by the way.)
Different dialects use different words for relations: for example, in my Grandma's village (not far from Khodoriv), maternal aunt is called «ву́йна», paternal aunt is «стри́йна», and «те́та» is used for distant female relatives of older generations for which we have no other word. Different dialects just use the words differently. However, there is a single version of language that is understood throughout Ukraine, the literary Ukrainian. The literary Ukrainian uses «ті́тка», and this word is understood throughout Ukraine. So this is why this word is taught here.
It's used by Ukrainians in 1920s, and nowadays you can hear it sometimes in West Ukraine