I want to reset the old carrier settings under "settings -> general -> about" and remove a carrier button which was added by my old carrier under "settings -> phone". Is this possible?
Thank you very much in advance!
You should be able to make the iPhone download the new carrier settings by tapping Settings > General > About, if it's a partner carrier.
See this Apple doc for more info -> iOS: Updating your carrier settings
Note: If you insert a new SIM or micro-SIM card into your iPhone or iPad, you'll need to download the carrier settings for the new carrier wirelessly. Carrier settings are available only for iPhone support carriers and iPad partner carriers. If you're not using a partner carrier, you will need to set up the carrier's settings manually. Refer to your carrier for appropriate settings.
Edit: What carrier are you trying to switch to? It sounds like you'll have to update the settings manually.