Can I remotely control an iPhone from a Windows laptop? Does TeamViewer Quicksupport only let me  see her  iPhone screen from my Windows laptop? Between computers, TeamViewer lets me control and operate the other computer.

This YouTube video and article refers only to screen sharing.
Yes, that is correct. The current version of TeamViewer for iOS only supports sharing the screen, not remote control.

Other solutions makes remote control possible, but they require the individual app you want to control to support that remote control solution.