Fix various editor state related regressions

[KoBeWi]

Follow-up to #116905

• Fixed dummy scene state overwriting the first scene (fixes Godot does not save editor settings #119442)
• Fixed scene tabs not updating on next/prev tab (fixes [4.7] Selected tab not updating when Ctrl+Tab cycling through open scenes #119470)
• Renamed selected_nodes state key to $selected_nodes
  • Avoids errors after upgrading (see Overhaul scene loading and edit state management #116905 (comment))
  • The new name makes more sense, because other keys are editor plugin names. This one is special
• Loosely related cleanup of one method

• Bugsquad edit: Fixes Can't use get_node() with absolute paths from outside the active scene tree. on editor startup #119492

[bruvzg]

Fixes 119442.
Fixed 119470 (note for testing, only the leftmost scene was not loading settings).

The rest of changes also look good.

[kleonc]

Tested the artifact, fixes #119492.

[Repiteo]

Thanks!