Should I let my cats be around each other?

My mom got a kitten a couple weeks ago and she's about  12 weeks old now and I have a cat named Owl who's 7 months old. Before when we got her he did not want to go near the door leading to where she is and now he does. My question is, should I let them be around each other? I'm nervous to take it to the next step because I don't know how Owl will react. I'm also nervous because she sticks her paws under the door and has gotten him with her claws once before. I know when I do I should have blankets/towels on hand just in case. If anyone has any advice I would greatly appreciate it

There's no reason why they shouldn't be on opposite sides of a closed door, provided that there's no anxiety or aggression from either of them. In fact, feeding on opposite sides of a closed door is a great way to create a positive association between the two of them, and is one of the first steps on the road to introductions. Ideally you'd start with scent swapping using cloths, and only move on to the feedings when they're comfortable with those cloths. However, there's no reason why you couldn't set both processes in motion at the same time if both cats are amenable. Kittens will often adjust to newcomers far faster than older cats do - they might just be letting you know that they're ready to get to know each other.

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The reason they were separated was because my cat would hiss at her when he saw her and now hes not hissing anymore.

The fact that Owl has stopped hissing is a great sign. It shows he's ready to start getting to know this little girl. Just keep in mind that integrating cats is a process, and can easily take weeks or months. Have a look at the articles I linked to. They set out the best process for stress free introductions for all concerned.