It's an SE rule privileges are strictly bound to the users reputations (mods being an exception):
If your new reputation brings you below the requirement for any privileges, you will lose access to those privileges
Badges once awarded are to be kept by the user and (almost) never taken aways they stay with the user. Except for tag badges.
Most badges a user can earn don't have any effect on site functionality; they are simply signs of accomplishment and bragging rights. A user's abilities are governed not by his badges, but by his reputation.
The Stack Exchange administration has stated repeatedly that "regular" badges never go away unless they were obtained by heinous cheating. Behavior that qualifies as "heinous" is defined by devs on a case-by-case basis
As for the massive unupvote you have garnered, it definitely falls under the category of serial upvoting and is against the rules and is definitely wrong:
The votes are un-upvotes, but they are voting behavior non the less. And they are targeted at specific users, not the content.
As such this goes against the rules - it is not allowed in any shape or form.
If the answers again gain 10 votes, will I earn another nice-answer badge for the same answer?
No. It doesn't. Speaking through personal experience.
So, to stop users from playing games with vote-unvote-vote-unvote, why not enact a -1 for un-upvote too?
This, would be outrageously wrong! Imagine X gave an bad answer and Y downvoted it. The vote got locked after 5mins (preventing unupvote). Then the answer got edited (by X or someone else). The edit actually improved the answer to a level that now it doesn't deserve a downvote but isn't good enough to deserve an upvote. So Y decided to remove the downvote. Handing Y a -1 for that what be wrong altogether. That is the very reason the vote are unlocked after the post is edited because editing could make the post bad or it could make it better and that's why Y is given the right to reconsider his/her vote.
What's bad is abusing this system