I'm not as knowledgable as other members such as rridgely, Andavari, Tarun etc. so my answers are from my personal experience.
I wouldn't say that Housecall is better than Ad-Aware. I've noticed many people have different results than other other people using the same products. On my PC Ad-Aware hardly ever found anything other than cookies, which CC removes when you run it. MS AntiSpy never seemed to find anything on my PC and locks up quite a bit. Other people thank it's the best one to use and I do use its Real Time Protection feature. Some people have a favorite scanner and don't use any others. I use quite a few different ones and run one every couple days so that they overlap. It's pretty much a matter of personal preference of how they work on your PC and which features you like best. One thing is critical to either method which is to check for updates frequently. Malware authors are always trying to outsmart the AV scanners and find new ways to get into your PC.
I would try to run Housecall before doing anything with Ad-Aware to see if it works. The advice to remove Ad-Aware is from Trendmicro's web site. It says Ad-Aware has a feature that prevents Housecall from updating files it needs to run when using Firefox. I think IE works with or without Ad-Aware.
The difference between using IE verses Firefox is that IE can use ActiveX or Java whereas Firefox only works with Java. If you use the ActiveX version you may have to give Housecall permission to install the ActiveX control and with Java you may have to update to the newest version. Housecall will let you know either way.
This link is to
Housecall 6.0 which is an older version for IE only that is quick and easy to use.
This link is for
Housecall 6.5 which is the newest version. It allows you to choose either IE or Firefox. It's a little more complicated (more features and scanning options) and it takes quite a bit longer but seems more comprehensive than 6.0.
If you're pretty consistent with how often you scan and update, after a little while the scanners won't find malware very often which of course a good thing!
Good Luck, Happy New Year, and if you have any questions don't hesitate to ask!