I am familiar with the diaresis, more common I think in British than American usage. The most common use I have seen is in "cooperation". I wouldn't use it myself in preparing an e-book, preferring "co-operation". However, if the original text uses it, what are you to do?
I just use the extended character set, select the vowel with the umlaut, and hope for the best. I'd be interested to see which character sets would produce a strange result.
Mind, I always use Calibri through clunky Microsoft Word, so I'm not sure this has been a problem to affect me.