For the main menu, I prefer Option 1 as the first choice, then Option 2 as the second one. The second option looks nice albeit not that functional (because of side scrolling which may be a bad design choice, I don't know I'm not a mobile app designer), while the first option is better and more familiar to mobile users.
About the modal, I approve 100% no further comment.
About saving, I also approve 100%. Just make a cue to notify the user that the setting has been applied and that the user can close the window.
About MobileOptions and Preferences, I want only one preferences tab (or settings, the label doesn't matter), and not two.
Even though I don't access the preferences that often, even on my laptop, I just want a similar experience in both mobile and desktop versions.