As long as the mains is plugged in before you turn on, but then it wouldn't turn on anyway.
No, removing the battery should not cause any freezing or other issues. The only time there might be an issue is if there was a mains power cut. With the battery in it would take over, but with the battery out the system would turn off instantly and that could cause instability at the next reboot.
I think you need to look at other causes for this freezing.
This may be a strange quesiton. The laptop(Toshiba Satellite 600D-1GB) is new (purchased last December). I use it at home as it's large and heavy so I have taken out the battery to, switching on the plug when I use it. Then the computer sometimes freezes when starting up and I'm wondering if this is to do with having no battery installed or with some other problem. Should I put back the battery?