I'm no expert but I know for a fact that the Caucasian region has HUNDREDS of languages and it would not be a surprise to me if Khabib's posse consists of multiple non-Avars so they all communicate in Russian - a common second language they share. In fact, I think there must be a few Chechens in there who have their own separate language.
Khabib's accent also does not strike me as unusual - most Caucasian Russian-speakers have a strong accent but remain perfectly understandable to me. Each has his own "flavor" and I can more or less discern a Georgian from an Armenian on a good day but Dagestanis all sound the same to me.
And if you're interested in how the general Russian public views him... well, reading comments on Russian sports website I get the impression that he's more hated there than on Sherdog, tho the haters use the same kind of arguments against him - Muslim fundamentalist, Kadyrov's pal, bigot, hypocrite, not representing Russia properly (as he does not carry the Russian flag after fights) etc.
I personally liked Khabib right up to the Conor brawl. After that he had a chance to apologize and take a high road but instead got full of himself when Putin sorta gave his nod of approval for this "retaliation" - which left a bad taste in my mouth. IMO Khabib is an amazing fight and I love watching his aggressive groundwork AND I think that MMA is a great way out of poverty and conflict for people from poor Dagestani families and troubled regions in general... but they often keep a specific mentality and culture that is sometimes hard to understand or accept.