The difference between a 10" 60T and 80T saw blade is only slightly noticeable. In practice, chip out with either isn't a huge concern, provided you go slow. An 80T blade is naturally going to slow you down more, so that may be useful if your tendency is to yank the radial arm down and go.
Another consideration is the end grain of the piece, some trees are just flakier than others, in that case an 80T is a safer bet.
Teeth aren't the whole story. If you throw a crosscut table blade on a miter saw it might have a tendency to feed faster than you want. If you get a miter blade then there's not a ton of difference. The miter saw blade teeth are back angled a bit to avoid pulling the arm in on you.
So the safe bet is the 80T miter blade, but I'd be surprised if a 60T gave you any trouble.
It's burning wood you smell. If it's not a new saw, the issue is likely that the blade is dull. Time for a new blade.
If it's a new saw, it could be that you're cutting too fast through too hard of wood.
Best Answer
The diameter is important, as that's the maximum size blade that will physically fit in the saw. The arbor size is important, because that's the size hole that will fit on the saws mandrel. If either of these sizes are wrong, the blade simply will not fit on the saw.
The thickness of the blade doesn't typically matter, as far as fitting on the machine. Unless you're talking about a dado stack, but you'd not be putting that on a miter saw anyway. The blade thickness, or kerf, only tells you how much material will be removed when cutting.
The number of teeth influences the speed of the cut, as well as the smoothness of the cut. Generally, more teeth give a smoother cut.