By the way I am using Adobe reader X, the newest version.
I have the same Problem.
I´m using Adobe Reader X (10.1.3) with OS X Lion.
If I copy a text with German Umlauts into a HTML-Editor the German 'ü' is diplayed obviously ok and the browser shows it correctly too, but when I move the cursor through the text the German ü are two overlapping letter. Normally it should be one letter, namely the same as entering an ü. (Yes, the charset of my HTML-file is UTF-8 and the file is saved in UTF-8). The first of the two overlapping letters is a "normal" 'u' and the second is the 'COMBINING DIAERESIS' (U+0308) (http://www.fileformat.info/info/unicode/char/308/index.htm).
Before I switched to OS X I used Adobe Reader with Windows 7 and I never had this problem. Strangely the problem only appears with 'ü's. 'Ä's and 'Ö's are copied correctly...
Right now, I suppose there is a bug/feature/whatever in Adobe Reader X using OS X....
Did you find a solution meanwhile?