iPhone 4s is giving me trouble, can someone help me troubleshoot?
You have two different issues:

For the Hotmail issue:


Since we don't want to use this account and use the new one, they are still registered with the old Apple ID. To get them with the new Apple ID, you will need to delete the app and redownload the app with the new Apple ID


For the wrong email entered issue:


Go to appleid.apple.com
Choose manage my account
Sign in with the email as it was typed in incorrectly
It will say it is not verified and will give you an edit option on the side of the email address
Type in the correct email
Verify your account for the new Apple ID


I would correct the issue with the new Apple ID first. Then: 


Go to Settings> iTunes and App Store option.
Tap on the Apple ID logged in
Choose sign out
Sign in with the corrected Apple ID


Then delete the apps and redownload with the new Apple ID