On beta sites, anyone with 125 reputation can downvote posts. Anyone. Downvoting and commenting are hardly moderator-only abilities.
Or, to put it another way, everone on the site is a moderator, and downvoting and commenting very much are moderator abilities.
Instead of just jumping to conclusions about who downvoted you and why, try to figure out how you can improve your answer (see help centre page on "How can I write a good answer?", as well as the relevant meta post "How do I write a good answer to a question?" for advice). The onus is always on the original poster to write a good post in the first place, but what voters consider good is not necessarily the same thing as what the poster considered good.
As for the rest of your questions:
- No, moderators do not get paid. They are volunteering their time like pretty much everyone else on the site.
- Community-run moderator elections are unlikely to happen until the community demonstrates itself healthy enough to participate in them (i.e. graduates from beta); this is far from the case right now.
- Moderator reappointment is unlikely unless there is a compelling need for the Community Team to do so (e.g. if an existing moderator voluntarily resigns or is involved in blatant abuse).
The moderators were chosen directly by the Stack Exchange Community Team, but for the most part, their powers are the same as those available to any other community user (with enough reputation), simply accelerated. When it comes to deciding how to curate the site, they are users like any other with their own opinions and making their own judgements.
If you find yourself at odds with a moderator decision, or for that matter a decision by any community user, fine. That happens. It's a side-effect of being in a community made up of different people, after all. But moderators are expected to lead by example; if you disagree with the example a moderator is setting, you still need to ask yourself if what you're trying to do on the site is actually what the site was intended for.
Immediately calling for moderator replacement just because they "might" be tired (and because someone downvoted you) is unlikely to be taken seriously by the Community Team.