You are perfectly correct on all your assumptions:

• the follow-up = le suivi
• to follow up on something = donner suite à quelque chose¹
• to follow (somebody) = suivre (quelqu’un)
• to come after (an event) = suivre (un événement), succéder à (un événement) (more litt., probably not for all contexts)

Votre compréhension est juste :

• the follow-up = le suivi
• to follow up on something = donner suite à quelque chose
• to follow (somebody) = suivre (quelqu’un)
• to come after (an event) = suivre (un événement), succéder à (un événement) (plus litéraire, probablement pas adapté à tous les contextes)

_{¹ Please note quelque chose is written without a dash.}

_{This answers is for France. I’ll leave it to someone from Québec to answer if the idioms are different for this translation.}

I think the answer will change depending on the context and the timespan between the events :

• For “Following our meeting…’, you would say : “Suite à notre réunion”
• For “As a follow-up of our meeting”, you would say : “Pour donner suite à notre réunion” (you won’t say : “Pour suivi de …”, even if it is grammatically correct)

The French verb “suivre” is the equivalent of the verb “to follow”. The way you use the verb will change depending on the postposition you are using in the English version.

I also regularly use “faire un suivi” for “to follow up”.