i've owned leather chairs for years. while true that they can get hot this normally is not an issue unless you are in a house without air conditioning in the summertime. mesh chairs just do not have the comfort value padded leather does.
with that said, you definitely want to try and get a chair which has at a bare minimum thick braces which have multiple connection points on the plywood base and back. preferably you would get a model which doesnt screw into the sides but i havent seen many like this in your budget range.
i never had any issues besides leather peeling (we have cats which like to perch on the chair tops) when gaming but since i started using chairs for watching movies (mainly reclining with my feet up on the desk) i started having problems with the screws pulling out of the plywood sides.
for general use and gaming purposes though, just about anything should work. i would highly suggest going down to the local staples, officemax, furniture store or other destination to personally try out chairs. anything around $120-150 should be more than sufficient (i like buying $250 chairs which go on sale for $150 once in awhile).
just some ideas. if you arent in a huge hurry i'd wait for a sale.