Not sure when exactly, probably automatically after reaching a certain post count, maybe account age also matters. You don't need to do anything special or apply anywhere, just be a normal user of the forum.
Chat and conversations are almost the same thing.
The top bar have a chat icon at some point, it's hard to miss. It will also have a search bar/magnifying glass at some point. Conversation is a bit more hidden, but it will be left to the notification icon (at least on mobile).
As mentioned above, when clicking a username, you should see four options: Follow, Ignore, Start chat, Start conversation.
Not sure if the two last options are hidden for you, I can theoretically message new members, but they probably won't see it until they unlock chat/conversations, or maybe there is an error when trying to send it.