Unless it is an old mac (pre 2008 or so), surely what you are calling a ‘mini vga’ is actually a mini-dp?
If so, you want a mini displayport to hdmi adapter that goes straight into the mac and the tv, and don’t want to go to vga. They’re cheap, any old one will do (even the apple one is not ridiculously expensive).
I think you usually have to run sound separately, although on newer computers they can put sound down it.
if it really is an ancient mac, it doesn’t have dvi does it? Some of them had dvi, which is easy to convert to hdmi as well as the vga.