I just purchased two Thunder Power batteries for my foam airplanes. I just noticed the speed controllers on both planes use the thicker gauge wires to connect to the batteries. The batteries have the small gauge wires on them. Can I, or do I have to cut the connectors off the battery leads and solder a deans connector on it?

You can, you don't have to. strange question. strange answer too.

Do we assume that you already have male Deans Ultra on your ESCs? If you think that you will standardize to Deans Ultra... then, yes, replace whatever is on the TP packs with matching female Deans.... for small foamies Deans Ultra are probably overkill... if you aren't drawing more than 10A-12A you might be better with Deans Micro all around.

That was part of the problem. The ESC had nothing on the ends of the wires and they were the thicker wires on the Hacker X30 Pro. The battery had the smaller connector. I just put a Deans Ultra on the ESC and cut the connector off the battery, TP 1350mAh pack and put a Deans Ultra on that. I'm guessing that's OK.

It makes good sense to standardise on the type of connectors you use so that you don't have to carry adaptors around with your chargers. Even if Deans Ultras are overweight for your current model when you go to a larger model or at least one that uses more power you will have plenty of capacity in hand.

I standardised on 4mm gold (bullet) connectors way back which was fine to start with but now there is every size in 0.5mm increments down to 2mm and every ESC seems to be supplied with different sizes it has become a minefield! Too much choice can be a bad thing!
Simple and reliable is all we want.

Plan for the future and use a connector which will do everything you will be likely to ask of it. Come the day when you're cutting off connectors from your 2200s to put on your new pack of 5000s you'll thank yourself!