Do "thereafter", "ever after", and "ever since" mean the same?

Ever after and thereafter can modify a verb in the past tense, so we might say ‘They lived happily ever after’ or ‘They lived happily thereafter.’ Ever since is a little different, because one of the meanings of since is ‘until now from a particular time in the past’. That temporal specificity means that it must be used with the present perfect construction, so we’d say ‘They have lived happily ever since’ and not ‘They lived happily ever since’.

They mean roughly the same thing as one another, but:

• Ever since is used to mean after something in the past only (and not after something in the present or future, as the other terms are). (Here, I mean past relative to the speaker, not to the present: "Thirty years from now, it will have been twenty years since my wedding and I will have been happy ever since" is fine.)
• Thereafter doesn't necessarily imply a span of time after an event (though it can, as in the example you give in the question): rather, it can imply an instant of time after an event. Thus, I can walk into a room, and John can walk into it "thereafter" (but not "ever since" or "ever after"); on the other hand, I can walk into a room and be in that room "ever since" (or "ever after"), meaning from my entry until now. In fact, even where thereafter does mean a span of time, as in "They married and lived happily thereafter", it doesn't mean a span of time until now necessarily (they may have lived happily thereafter for a year and then miserably), whereas "They married and lived happily ever after/since" means until now.

That's my impression, anyway. See also Barrie England's good answer regarding choice of tense/aspect.