Do upvotes and downvotes affect my reputation in the Mathematics Stack exchange site from this Mathematics Meta website? Also do comments or anything that I do on the meta allow me to earn badges or privileges on the real http://math.stackexchange.com? Any help would be truly greatly appreciated. Thanks so much in advance.
As far as your questions about badges is concerned, you do not get badges on main for your activity on meta. There are two exceptions:
However, you can earn badges on meta. (You have probably noticed that as you have already earned some meta-badges.)
Upvotes/downvotes on meta do not influence your reputation on the main site. See also here: