This is a duplicate of Mac Terminal Copy on select like Putty but I would like a solution for Linux.
So basically, (how) can I set the selected text to be copied and pasted on right click?
As already pointed out by Alex, it is , at least, not a good thing make right click act like paste command. On terminal, you can usually set this up in preferences. Alternatively you can use terminator instead of default terminal and get this feature along with a lot more.
You can use the middle mouse button, I use this all the time on GNOME based desktops, might work on others however I haven't tried.
When you highlight some text it gets copied in the middle mouse clipboard, then when you want to paste what you have highlighted you press the middle mouse button.
The middle mouse butten clipboard does not overwrite what you have in the normal right click/menu - edit paste clipboard so you can store different things in both clipboards. I believe the normal clipboard does overwrite what you have the middle mouse clipboard when you copy things in to it.