Question re. Mail Count Badge with Multiple Accounts

I currently have two email accounts configured on my iPhone, both of which are Exchange accounts. One of them is my work email and the other is my personal email. While I'd like the mail count badge to reflect the number of new messages I have in my personal email address, I'd prefer that my new work emails not contribute to the count. In other words, I'd like to have the mail count badge reflect only new messages in my personal email account and not my work account.

I've looked through the settings and I haven't been able to find any way to exclude an account from the new message count, so I don't know if it's possible to do this. If anyone might be able to help, or if anyone knows for sure that this can't be done, please let me know.

Also, on a somewhat related note, is it possible to rename an email account, or is the inbox/account name displayed on the mailboxes page set to whatever name was assigned when the account was created?

Thanks to anyone who might be able to help!