First of all, I clarify that I don’t use AVAST! and therefore do not know the definite answer. Nevertheless, for your guidance, I provide my indirect but related knowledge and experience on the subject.
In theory, as already commented, there should normally be no compatibility problem of an antivirus with any of the tools you mentioned, as none of them is a proper antivirus.
However, when I recently updated my AVIRA to the latest version, it detected other tools and advised me to uninstall: Ad-Aware, Spybot S&D, Superantispyware and SpywareBlaster, because of compatibility problems – strangely it left out MBAM. This is in contrast to previous updates which found no such problem. It’s also in contrast to the claims of the tools named that they have no compatibility issues with any major antivirus.
However again, I ignored the AVIRA advise, kept all tools named and do not seem to experience any problems so far. I was also told by one of the tools that antivirus tools just want to cover themselves in giving such advice. It’s a kind of “false positive”.
Conclusion: AVAST! are the people to tell you, but even if they inform of incompatibility, it does not necessarily mean there is one.