Neon 5.13
I'm wondering is it possible to use a full-screen launcher of default launcher as a launcher in gnome-shell:
- If the app was not launched yet - launch it
- If the window was minimized or has no focus - it brings this window to the front and focuses on it.
Tried to google the correct way of doing it, but have failed.
Thanks!

To switch to the fullscreen launcher: right-click on launcher icon -> Alternatives -> select Application Dashboard.

As for having it open already opened applications, I tried a number of launchers for Plasma 5 but didn't find one that would do it.
However, there are a couple of other options you may consider:
* the text-based launcher Krunner (alt+F2) will match windows, allowing you to either select already opened windows or launch new instances.
* Icons in the taskbar can be pinned through the right-click menus to show a launcher icon even when not open.

(also, some apps like browsers can be configured to not open new windows when launched)