My first thought - if it works in one browser but not another, it might be a cookie, or a cache issue (I've had both affect me on multiple occasions in both IE and FF).
- Column size issue
However, if you've tried deleting both cookies and cache and your module is moving over, one thing it could be is the way the browser renders the width of the search box. In your admincp, one of the options is the width of your Google search box, this is a global setting for your drop-down, the CMPS portal search, and for the search page. Try tweaking this variable smaller than what you have as IE and FF might render these different sizes.
- Drop down not working
One thing this will not do on a default board is disable javascript. That isn't to say that this modification will or will not have issues with another modification. If the drop down does not work for guests but does for members, this makes me think back to a cookie/cache issue.
So, ultimately, I would have to either say A) its either a cookie/cache issue with one or more browsers or B) an issue with a mod conflicting with this one - or a very slim chance that it is both, but unlikely
Hope that helps.