.mbs is just a standard UNIX mailbox format, any program which would allow you to import saved mail should be able to read it. Having said that, I haven't looked at enough Android mail programs to say which ones might have such a feature.
If you were signed into sync before you can try cleaning the app data on the phone and try starting again. You can then sign in to sync, and you should be able to see your previous speed-dials by clicking the sync icon in the bottom-bar. When you are there you can select "Import all" to transfer them to the speed-dial page.
Hard to say what's cooking, for some reason Opera is faster now but I've got some connectivity problems with Google servers so perhaps something is wrong on their side. Afaik data management doesn't depend on browsers.
Sorry to hear about your issues. We are working on improving password management. We will make an announcement in the forums when we got a solution for this.