When you print on that printer/paper, yes.
Unsynchronized means that your Photoshop, Illy, whatever do not have the same colour settings.
The answer to your first question is no, you choose the profile your intend to work with for that type of file file, for print it is usually Adobe RGB 1998
or Color match RGB.
You choose your paper profile type in your Printer driver under paper type.
Do not choose it in your color settings which is a print job specific profile. That intended only for output.
If you are working for the web choose sRGB.
As far as your second question it means you have other Adobe applications installed as pat of a suite or master collection and they do not match.
What you do is using say photoshop you select your rgb cmyk and grayscale profiles and save giving it a distinctive name the you launch the Bridge and then go to Edit>Color Settings and select that setting or one of the presets settings and then you can select to apply that settings to all your applications from the Bridge with one click.
Jacob does not have any of the CS programs or suites and is not familiar with the process.
I am surprised at his answer in general but he is quite mistaken.
If you want to use this profile for your printer and paper or a custom one then do it in the print dialog.
Sorry: less pressed by another task (which is still pressing, though less) I would have answered more clearly and fully:
When If you will only print on that printer/paper, yes you may (I felt a bit uncomfortable about keeping to my general effort of being terse).
Unsynchronized means that your Photoshop, Illy, whatever do not have the same colour settings (this is the answer to the question, though not specifying the solution (not everyone uses Bridge as far as I know)).
For some reason, or none, the colouring option is off (the bold italics should have been red).