Can markers be bound to parts or clips, so they move together?

Did you check on menu Edit Range Cut Time - and see if the shortcut you use is assigned to that function?
If there is a mixup in keyboard shortcuts from possible earlier version you have or similar - use menu to be sure.

I think you are using range selection tool - that is not what you are to use.
Just drag a range in upper part of ruler, then do you cut time or delete time, as needed.

When I do as you do - I don’t get a selection between locators in ruler bar.

You can configure all track panel controls and save as presets to use for various needs while you work. Bottom of track view can select settings and such a preset to liking. One of the things Steinberg did really well I think.