I'm guessing you're not too familiar with snake teeth. No doubt your snake does have a normal mouth full of teeth. If you actually had a snake with no teeth you'd have some sort of extreme problem. If you're sufficiently inclined you can inspect the mouth (using the edge of something like a credit card can open the mouth and allow you to take a look). Their teeth are not like human teeth or crocodile teeth which stick right out, they're small, fine and curved backwards and may not be obvious to a novice observer. Stick your finger in the snake's mouth if you need further confirmation that the teeth are not there (if the snake actually had no teeth this wouldn't be a problem for your finger, and your finger would slide right out of the mouth).
Do a web search for some small python mouths to get an idea of what they look like. You'll see plenty of pictures which show that the teeth aren't always obvious, but your Diamond does have about 100 little teeth.
The feeding issue is a common one, there's not enough information here for me to comment, but there's no shortage of discussion and advice on this topic.