It looks like I had Malware.Trace on my computer (see attachment) which MBAM removed easily. I rebooted as instructed and the behaviors mentioned above went away. MBAM doesn't find anything now.
I also had Adware.InstallBrain in a file being downloaded from the Internet but it looks like it was stopped mid-download and blocked.
After some reading on the internet, I think I will continue to run Avast! free edition + MBAM free version and quick scan monthly with MBAM.
Thanks for all of your help.