English question, please help.?

They both are correct depending upon your intent. If you are referring to a single collective memory about a single incident then the first is used. However, if you are addressing multiple memories about multiple incidents or for a different point of view about a single event then the second is used. The semantics about whether a group of individuals can have exactly the same memory about an incident is what clouds the issue here. If your favourite team lost a game then the collective memory of the fans present is that the team lost so if you wish to take an action to remind the fans of that loss you can use "I want to refresh their memory"; I hung the uniform of the opposition team on wall of our teams change room for the players to see because “I wanted to refresh their memory that last time we lost”.

It depends on whether or not you're talking about more than one person. If you are talking about one person of an unknown gender, the first sentence is correct. If you are talking about two are more people, the second sentence is correct.

The second one is correct because "their" is plural so it requires a plural object.