As a new beek, and if I think they are OK, when is it good to open and check to the hive.
The safe answer is that you can do a deep inspection when you have drones or access to queens. If you have someone in your area that is experienced, check with them.
When we were in MS with Bud, we split hives in April. I could never do that where I live! I get snow in April.
You should be able to gauge how many new bees are coming out by orientation flights, but that does take a little experience. A flashlight and a look down the frames can be helpful. I like to add my new brood boxes under my overwinter box, and my honey supers on top, but that's just me. I know a lot of people don't do it that way.
The other thing to remember is that swarming is natural. Nice if you can avoid it, but if they decide they are going, they are going. In the long run, it's usually not a big setback for the hive. You get a new queen and if she's a good one she'll more than make up the numbers.