1. I have never heard of that name / heard that name
I could say that "never heard of" means "never had any indication that that name exists", and "never heard" means "I've not heard that name spoken". But maybe this is inventing a difference that isn't there in normal speech. I think most people would mean the same by both phrases: "I haven't come across that name before, in any form".
2. Please put it on my tab.
This is fine.
3. It must not be his first time in the town, right? So, he would know his way around here.
This sounds alright, but I think "It cannot/can't be his first time in town, right? So,Ö" might be slightly more natural.
not a teacher