Attempting to get a similar User Experience on multiple devices

Hello Stackideas Team

I have 2 templates installed: Vanilla and the ES Mobile Template.

1) The home button (House Symbol top left)

MultipleDevices01-HOME-Button

On a desktop clicking home takes me to the home page:
https://finalbug.net/

On a mobile clicking home takes me to the stream page:
https://finalbug.net/stream-social-learning

I have a vague recollection of being able to set the home button to go to either home or stream. In my case I would like the home button to go to the home page in both cases. I guess that I have to set this in either one or both templates. Kindly advise?

2) The mini Menu on Desktops

I am not sure what this is called but you can see it in the following screen shots:

MultipleDevices02-MAIN-TOP-USERPROFILE-Login
MultipleDevices02-MAIN-TOP-USERPROFILE-Register

Where is this in the back end? Where can I turn it on and off?
Does this appear only on desktops?
I guess so because it is a Vanilla feature? Correct?

3) Once logged in on a mobile there is a little Cog Symbol down bottom right.

Understanding-Menus03-Cog-Bottom-Right
Understanding-Menus03-Shows-More-User-Stuff

Is this the same menu as in this screen shot

Understanding-Menus03-Desktop-Other-Profile-Stuff

Are these the same menus? Just for different devices?

4) The menu with the APP like large buttons.

At one point maybe about 6 months ago on a mobile phone. (ES mobile template)

I clicked on the the “MENU” and saw a screen with nice big buttons
(Please see the BLAH screen shot)
Is this now gone?
Can I switch it on?
Is it a setting some where?

kindly advise

Paul