I'm not sure what the printer and AVAST have to do with Internet Explorer, but if I understand it correctly, they're just using IE as a generic term for a web browser. Unless either of them specifically use ActiveX, you should be able to use any browser.
As for Spybot, you probably won't need to use it if you switch to something like Firefox. Most spyware today comes in the form of ActiveX or tracking cookies. Firefox doesn't support ActiveX, so that's out of the question, and if you enable cookie-whitelisting, you won't be affected by tracking cookies either.
EDIT: Just noticed you mentioned you wanted to use Avant. Well, since Avant is just a front-end to IE, you should be able to use it without problems. It will support ActiveX, so the printer/Avast/Spybot shouldn't have a problem if any of them need that either. Your best bet is to install Avant as see if everything works. If it doesn't, you can always uninstall Avant and go back to using IE anytime.