hmm. off the top of my head: 5 centimeters per second, Steins;Gate, Fullmetal Alchemist: Brotherhood, Hajime no Ippo, and maybe anohana (though I recently finished it so it might be just some recency bias)
Has anyone ever said something to you that set you on a path to prove them wrong?
Hmm I guess in small ways, mostly in arguments I think. Outside of moment to moment things I don't really find myself being vindictive or out to prove something against someone.
I didn't know that you couldn't have multiple accounts because i didn't read the rules. How long will I get restricted for?
3 months is the current minimum wait before appealing a restriction. If you are currently not restricted and genuinely did not know the rule at the time and have not made any egregious misconducts with the accounts, you can send an email to accounts@ppy.sh explaining your situation and your now awareness that having multiple accounts is against the rules. YMMV though, so do be careful.
Be active in chat and on forums, report users (but don't overdo it), be well behaving, have altruistic reasons for wanting to become a GMT member, and fill out the application. Bonus if you speak another language whose channel could use more moderators, and/or if you have some other outstanding qualities. And then good luck.
Varies from person to person. Most people are likely to improve more quickly by using a tablet. But without knowing how good you are currently it's kind of hard to say lol