My main account dream_miner has been banned due to private message spam. Is there any banning section that accepts appeals?
Yes, the Meta section.
While being banned, you're not allowed to post outside of it, and only allowed to post in the Meta section in threads specifically appealing your ban.
This account is used to contact very important developers that i am working with.
That's ban evasion and will likely cause your ban to explode, and also prevent your ban from being lifted.
You're not allowed to PM/post/...in any way other than specified above.
One of another accounts of mine was banned for same reason but only for 7 days. Also the "7-day" banned account is Newbie, while main acc is Member.
That account wouldn't be still banned aswell, would it? Posting from your "main" account while your newbie "another" account is banned once again is ban evasion and a reason for all your accounts to be banned if discovered.
You may just need to start over with your new account.
Are you suggesting that the user evades their ban?
It's not the account that's banned, it's the person. All accounts in their possession, future and present, would be banned aswell.