This may well be a "newbie" question: I am for the most part quite familiar with the SE platform via my participation over the past 9 months at MO. But their meta site is totally different from this one.
MO is using SE 1.0 platform. math.SE is using SE 2.0. So your experience in MO cannot be completely applied here. In particular, there is no default meta site in SE 1.0. MO's meta site is using third-party code (Vanilla 1.1.9 from Lussumo), while here we are using the SO backend.
I guess I don't understand why reputation is displayed on the meta site at all. It seems to create the impression that you gain or lose reputation based on your posts here.
You should ask on https://meta.stackoverflow.com/. I don't know the specific reason. But by definition “Reputation is a rough measurement of how much the community trusts you.”, it should give an impression on
how much weight you should put on one's argument how involved the user is on the parent site.
Also, why is it only synchronized hourly with the main site? Would it be problematic to have the updates be instantaneous?
The meta site and the main site do not share database the user info database. Thus to display the reputation they must be synced. And to reduce the server load, the frequency is set to be not so high.
Further, when I go to my user page on this meta site, upvotes show up as "reputation gained" in exactly the same way as on the main site. Can someone explain why this is a feature and not a bug?
Again, questions about the non-site-specific SE 2.0 architecture should better be asked on https://meta.stackoverflow.com/.