I have no idea about the browser specific problems on your production machine. I would use Firebug and/or Chrome dev tools to look at the network requests for any problems and the script console to see if any errors are happening, check the mojoportal log etc.
As far as removing the edit buttons, that is not possible because MyPage/WebParts are intended by design for user interaction to personalize the content by adding content they are interested in from the catalog of available content.
The images are hard coded like:
so the only way to customize them is to replace those files with new files of the same name and do the same again after upgrades.
I guess you could put a 1 pixel image for the edit icon so it would not be visible.
Note that as of recent source code, the MyPage feature has been moved into a separate project mojoPortal.Features.MyPage and moved out of the mojoPortal.Web project. And this project is not included in any of the default .sln files so you would need to add it to your custom .sln file if you want to work with the code.
WebParts never caught on in ASP.NET and are basically dead end technology so the MyPage feature is deprecated, it still exists and does what it does for those who want to use it but it is disabled by default in mojoPortal 126.96.36.199 (though it is included in the package and can be enabled by config). At some point in the near future I will stop including it in the package and make it a separate download. I have no future plans for WebParts or the MyPage feature, but one could fork the code of that project now that it is separate from mojoPortal core and customize it as they see fit.
Hope that helps,