By using system restore, it basically reverts the computer to a point in time (whatever day that you choose).
If avast was installed after the date that you restored to, avast won't really be there. Which is sort of the point, if you install an application that prevents the computer from working correctly, system restore will get it back to when it wasn't installed, thus fixing the problem.
In order to restore avast, just re-install it. As well as chrome and firefox. It shouldn't take much time to get those back up and running.
As for the password for your desktop computer, it would be easiest to ask the person that you purchased from to give you the passwords for it.
Gopher John is correct that you can try the administrator account, but you don't need to boot into safe mode to do it. When you get to the login screen with the usernames on it, just hit ctrl-alt-del twice.
You'll get a simple login screen where you can type a username and a password (if required).
However, most administrator accounts are disabled by system manufacturers, so I don't know if it will help.
I have several different password reset tools in my toolbox, this is one of them: http://home.eunet.no/pnordahl/ntpasswd/
It's a little difficult to use for novices, but it you take your time, and read what it's asking, it's pretty straight-forward. Use the program to reset a user, reboot, and login.
If you need more help, let us know. Welcome to the forum