Can I post HTML-formatted messages on this forum?
The short answer is yes, you can. Just take a closer look down at the comment entering form and you will notice the line "Format comments as" with choices available. When you enter some HTML tags and want them to actually behave like tags - use the HTML-formatting choice. Otherwise, all the HTML control data will be stripped from your post and you'll see your tags in plain text only. It is pretty much self-explanatory. But bear in mind that not all tags are allowed. Some ugly ones from pre-HTML4 days are banned for life. Some others are omitted as unnecessary or for security reasons. There's no need to list them all here, just remember that if some tag is not working, that will be why.

Is there a kind of private messaging for users on this forum where they can communicate directly without disturbing anyone else? How do I use it?
There is "whispering". Although it's not the same thing as private messaging on other forums, it works really well.
As the official Vanilla FAQ puts it:
"A whisper is a private message that appears inline with a discussion. Only you and the recipient of the whisper can see the whisper. It won’t affect the paging or comment counting of discussions. It’s a way for you to say something secretly to another member in the middle of a discussion. You can also create entirely private discussions by entering a whisper recipient when starting a discussion. Then only you and the recipient can see the entire discussion.
Whispers can only be seen by two people: the one whispering and the one being whispered to. You cannot whisper to multiple people at one time."