the Z10 supports Android apps and based on the issues found on the internet it may be the base app needs to be replaced.
If you already tried a few things the next may be to simply do a full backup of your information then do a factory restore to original levels . If the browser now works then there is a corruption somewhere and try loading your other apps one at a time then check.
I checked my Z10 both with BES and BIS service and both work.
This is the Facebook help link