They should be perfectly fine. I'd introduce them when you are going to be home and keep and eye on them, but I've never heard of it being a problem. (I have large African Cichlids where this is a big problem )

Ok, goldfish enthusiasts will tell you that a 10 gallon tank is too small for that many fish. They have told me that 1 goldfish for every 10 gallons. Meaning 2 goldfish 20 gallon tank.

That being said, I think that 2 gold fish and a betta in a 10 gallon tank is perfectly fine. I had 1 gold fish in a 5 gallon tank and when these people told me it was wrong I got a bigger tank and he died. Stick to what is working for you. I wouldn't add anymore though. Also if it's an un-cycled tank do 20% water changes for a while to lower the ammonia level until the "good" bacteria has a chance to grow. Even a cycled tank needs frequent water changes when new fish are added.

If you have any other questions you can ask me. In the past few years I’ve become a fish enthusiast myself.