I'm trying to add a friend in Steam but it doesn't allow me because my friend slots are full. I am trying to delete one of my friends to open a slot but it doesn't work. It gives an error saying that it failed to delete the friend.
Check if your Steam Family Sharing is active. Lot of functions is disabled while Family Sharing is enabled.
If isn't that you need to contact steam support.
Or just up a level to open 5 more friend slot.
I assume your account is limited?
This means you didn't bought something for at least 5$ in the steam market.
You can add no one and other things are limited. If your friend has got an unlimited one he can add you, because with a limited account you are still allowed to accept friend requests.
You can check here what you can do with a limited account and things you can not do.