I have always moved the window to another workspace using ctrl+alt+shift+ArrowKey.
But now with Ubuntu 17.10 I can't do this. And the worse part is, alt+TAB is tabbing through all the windows not just the windows open in this workspace.
I have compiz, I have tweaktools, I have tested all combinations from its documents, I have asked in #ubuntu irc
channel, nothing seems to be working.
You can install this extension to enable this.
NOTE:
Before you install the extension, you will need to install the GNOME Shell integration for your web browser and you will need to install the
chrome-gnome-shell
package usingsudo apt install chrome-gnome-shell
Then you need to flip the switch in the top right corner of this webpage.
To answer the first part:
ctrl-alt-shift
+up/down
works, but left and right don'tYou may use super+shift+page up/page down key combinations to move windows to workspace above/below (Source).
For the second part of the question see this: How do I make alt + tab cycle through individual windows only in current workspace in GNOME 3?