Why does the App Store prompt me for an old email address when I try to update apps?

What do you mean by 'want to make up dates' ? What exactly are you trying to do ?

All apps, and any other content downloaded from the iTunes stores, are tied to the account that downloaded them

I have this exact same issue. All apps 'purchased' under my account yet since my cousin logged onto my phone for something many months back, can't seem to disassociate his ID with my own and update any apps. Some apps update automatically in background but a number of them now seem tied to his apple ID and I can't get rid of it even though they were downloaded and 'owned' by me.

Have tried various suggestions from other posts such as signing out with my ID out of app store, going to app store and trying to download a new app/update existing but again it pops up with a request for his login credentials........very frustrating!

Would be very interested in hearing any suggestions on solving this?

If you are getting his id when trying to download updates to them then that implies that those copies were downloaded onto your phone by his account (or, if synced from a computer's iTunes, were downloaded onto that computer by his account). Have you tried deleting them and downloading them on your account ?

I have deleted all emails but the current one I use and the old one is still stuck in the sign in box. Not sure what to do next.

The sign in box when, when trying to download updates to an apps or buy new items ? If you've changed the primary email address on an account : Change your Apple ID - Apple Support

If it's showing when trying to download app updates then is it a different account ?