How to type an e with an ' over it or an o with two dots of it.

If you mean é and ö then it depends on whether or not your font and encode setting supports them. For me pressing the ALT-e key combination followed immediately by either e or E produce é or É. In the same manner pressing the ALT-u followed immediately by either o or O produce ö or Ö characters. (Alternatively, you can go through the "Font" menu, select the font actions button, and select the "diacritical" combination characters directly from the table window.)

It is the same as the "Option" key. (My keyboard uses both designations but "ALT" has fewer characters and I'm lazy.)

On mine it doesn't have to be done quickly. Just in succession. That is press the "e" key while holding down the "Option" key. The release the "Option" key and press either "e" or "E" key. (Actually, pressing any vowel will render an "accented" diacritic character -- e.g., á, é,í, ó, or ú.)

I held down the Option key and hit "E" twice quickly and I get just the ' but no e underneath it.

If you are pressing OPTION-e and OPTION-e then all you will end up with is an apostrophe. As stated above, press OPTION-e and e (no OPTION key for the second press). I tested this in iMovie '08 using the default font associated with the "Transparent - Black" title option (i.e., the Helvetica Neue font with default encoding for the application) and it works fine for me.