Is there a way to hide Chrome's "User Profile" and "Extensions" areas?

What you have circle as a single area, is actually two (or possibly three) areas:

• Extensions (None shown in your screenshot, but this could be useful for other users)
• Extensions Toolbar Menu
• Profiles

You can hide any extension icons by right-clicking on them and selecting "Hide in Chrome menu".

The 'puzzle piece' icon that you see there is the Extensions Toolbar Menu, you can disable is by searching for that term in chrome://flags and setting it to default or disabled.

However, the profile picture is not removable at this time. There used to be an option to hide it under chrome://flags, but that seems to have been an experimental feature and is no longer available.

That means, for logged in users it seems the best solution is to set your profile picture to a solid block that's the same colour as your Chrome Theme. For logged out users, there's only the default profile icon.