An useful site to bookmark for future use:
https://singularlabs.com/uninstallers/security-software/Ask if any other a/v software has ever been installed then removed, and then find the appropriate vendor removal tool to completely remove any remnants left behind.
[EDIT:] Sorry, overlooked your question about running avast! 24/7 or as only on-demand scanner:
http://www.bleepingcomputer.com/forums/t/186533/is-it-bad-to-run-multiple-antivirus-programs/ avast! is designed to run as a real-time scanner and not as a on-demand scanner. Which means avast! drivers are running all the time whether you turn off the file/web/mail scanners or leave them active. For a second-opinion scanner use ESET:
http://www.eset.com/us/online-scanner/ As with all a/v solutions, carefully check (google) any detections and quarantine instead of deleting. Deletion of a detected file is irreversible. You will not be able to get it back.
On demand scanners:
http://www.thewindowsclub.com/standalone-on-demand-antivirus-scannersA further explanation is required as well: Simply removing/uninstalling an antivirus program does not mean all active running drivers for that program will be removed when you uninstall it. Which is why the link to SingularLabs is provided.
avast will see the left-over drivers from the old a/v program and vice-versa, and the end result is a system slow-down due to the left over drivers from the old a/v. You'll want to provide a clean environment for avast! to operate in, and to do that you will need to use of one or more of the a/v removal tools given above.