Is there a global/worldwide database of dissertations? If not, how can I access French, German, Italian, etc. dissertations online?
I am not aware of an international database, but basically all german dissertations have to be submitted to the German National Library (Deutsche Nationalbibliothek), so if they exist in physical form they can be accessed at the libraries in Frankfurt and/or Leipzig. If they only exist in electronical form, they are still registered at the library. The following link is going to give you a list of all german dissertations that are freely available online: DNB search
The DNB also has a manual about searching for dissertations, however I could only find a german version that is available here
For French PhD thesis, the most important repository is TeL. Many university now make it mandatory to deposit one's thesis there, so it has a very good coverage for recent years.
Dutch doctoral theses can be found at:
For German dissertations it is often a requirement that the dissertation is "published", which used to mean published by a publisher, though many universities have made this requirement less strict to allow for the new possibilities of the internet. However, publishing with a publisher is still common practice in many disciplines. This means that many German dissertions are not downloadable from the internet. If you have a concrete dissertation you want to read it can help to contact the author directly, they may be willing to sent you a copy.
In Italy, all PhD dissertations should be sent to the Biblioteca nazionale centrale di Roma. It appears, however, that the dissertations can only be read there. The library has an online catalog which, however, doesn't seem to index the dissertations (from the library's website it appears that the PhD dissertations catalog is only in paper form).
For further information, I suggest you to contact the library directly. There is also an international interlibrary loan service (Prestito Interbibliotecario Internazionale).
There is also DART-Europe http://www.dart-europe.eu/basic-search.php which aggregates info about opely available European PhDs.