What part of the sentence implies "Tal's" place/home/etc. Is it because אצל is before the proper noun?
It's kind of problematic here because from the translation it does imply - as you wrote - a place. Though as Pumbush said, it is often used to express general possession.
Etzel is used to show a place of possession or to indicate who's in charge . Shel is used in the same way as the word "of" or possession of a thing . הייתי אצל דויד. הספר אצל טל (the book is at tal's) הספר של טל(the book is tals הכוס של מיץ .