Re: How can I delete my emails from my iphone 5s BUT NOT delete them from my computer
It all depends on the type of mail account. If you are on a POP3 account then removing them on the phone will not remove them from the server. If it is a IMAP or Exchange mail box then it is a two way sync so deletes will also sync.

🏼This, but! You can usually change that by logging in to your email on your browser and checking the settings.
For example, gmail let's you set that behavior, that if you erase on a mail app, it doesn't erase on the server, but archives them.
Thing is that if you erase or move an email on your iPhone and then check your computer mail app, chances are it will be moved too on your computer email client (call it outlook, mail or others.
Anyway, check your settings and apply your preferences.
Yeah, yeah, Sent from Tapatalk and all that stuff