Hi. Yes you can. There is no problems running office 2007 on windows 7. Is your office 2007 licenced for three computers? If it is you will be ok activating it as long as you havent done all three installs. If you have you will have to ring Microsoft to get it activated.
Most people only have the single license.
If you want, you can uninstall it from your old computer and replace it with something like open office if you are short on cash.
As far as compatibility, office 2007 will absolutely work with windows 7 lol.