When I open a text file from Dolphin (right click -> Open with Kate), Kate launches and displays one tab (Show Tabs is active). When I open another file from Dolphin, the second tab is opened to the left of the first tab. This behavior is very unusual for applications that support tabs, and I find it hard to get used to. I would expect a new tab to be opened to the right; instead every new tab shifts all existing tabs to the right.

Is there a way to adjust this?

Huh, that's odd.
I'm not aware of any such setting.
I'd suggest firing off a case to bugs.kde.org
(also, why do they implement their own tab bar rather than just using a standard tab bar component like everyone else)