Officially, it's not allowed. But TNT are simpletons and act wrongly against their own rules very often anyways, as you may already know. If you delete a side, it may even draw negative attention to you from TNT, and in the worst case - they'll misinterpret it as suspicious activity and chain every account you own, despite your good wishes.
I wouldn't think too much on having 5+ accounts; what you should do is stop logging into and ditch the sides you care for the least and just focus on the accounts you want the most. Avoid logging in/out of too many accounts from the same IP (I assume your home IP), and if you want, you can fake that some of the accounts are not yours, but a family member who shares the same internet connection. A lot of users do this, whether legit or not and just make a small public note in their user lookups. Just don't do this and shadily transfer stuff back/forth like you own all those accounts, or TNT will catch on. If you're already logged into the sides you don't want or have another internet connection/proxy, you can change the emails on those before leaving them alone; to avoid chaining account freezes in the future.
It's not necessary to use a proxy in your situation, though it would be smart. Using a proxy is simple, but if you don't have to, don't bother. They're best used when cheating on sides, not for mains or any accounts you want to keep safe. This is because proxy connections are slow, unreliable ("die"), and if it's a commonly used public proxy: other cheaters on Neopets that use it could get caught and end up chaining to your accounts you've used that specific proxy on too, in rare, but unlucky cases.
& yes, you need to check that a proxy is working (and undetectable as a proxy) before using it. What users often mean when a proxy "dies", is when it drops and cannot connect, or just disappears from the proxy list or being able to be found. They come and go all the time, don't expect to be able to wake up to the same proxy every morning. But again, proxies are not important in your situation right now, so don't overthink on them.