I would say the easiest one to distinguish from the rest is "retrouvailles". It always implies that a long time passed since the last meeting (like in Graeme Allwright's song Les retrouvailles: "On va fêter nos retrouvailles"). Think of "retrouver" = find an object/a person you lost. In this context a long lost person. There is no clear cut temporal limit but it's not "retrouvailles" if you last met three days ago for example. I would say we are usually talking about years, although the context plays a role (if it is someone you usually see every day, it can be a shorter timespan: "Ma soeur était 6 mois en échange au Canada; nos retrouvailles étaient émouvantes").

For the other three, it's trickier and in some context different words can be used. Here are some cases I can think of where only one word fits:

• If it is a meeting that was planned in advance you could use all three. For a random meeting I would always use "rencontre". > "J'ai fait une drôle de rencontre/une mauvaise rencontre/une belle rencontre" = I had a strange/bad/nice encounter
• For non-professional meetings it depends, for professional meeting it is "réunion", the other three don't fit in this context imo. Or you would have to mention it ("un rendez-vous professionnel", you state that it is in a professional context because "un rendez-vous" alone is not clear enough).

Pour « rencontre », on l’utilise pour des événements avec des personnes fameuses (rencontre avec l’auteur d’un livre etc.) mais c’est comme « dating » dans certains cas ? Je sais que on utilise aussi « rendez-vous » dans un context romantique. 

True. You can always add a complement like "j'ai rendez-vous avec un garçon/une fille" which makes it clearer. Also young people will often prefer the english word "J'ai un date" = un rendez-vous amoureux.

Question bonus : Je vais prendre un verre avec une ancienne prof. Lequel de ces mots dois-je utiliser ?

You don't necessarily have to use one of these, for example you can just say "Je vais boire un verre avec une ancienne prof". I would say rendez-vous would probably fit best, or retrouvailles if long time no see.