I send you installation instructions.
All domains can have own menus as usual (or share all menus from other site).
All domains can have own modules as usual (or share all modules from other site).
Ability to add custom CSS files for a domain will be added soon.

... if it allows you to configure custom CSS filename - it is fine, template settings will be separated (each site has own template settings as usual), so each site will load own .css file. Otherwise we can add an option for including of custom CSS files per domain in MightySites, it is not ...

You can install extensions as usual and use any templates.

New version is ready, editing of items doesn't use a modal popup-up window now but uses a usual window.
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All features are working as usual, JS integration will also work once JS is 3.0-compatible.

MightySites can help you to load users data from main site in all subsites, so users will be always same in all sites (any user-related action in any site will update database of main site). And single login/logout from 1 site to other sites is also possible.

Akeeba should work as usual.

FB Connect works, but if you configure single login in MightySites from site A to site B,C and user logs via Facebook in site A - it won't be auto-logged in sites b and C. He will be only logged in site A as usual.

I have sent you installation instructions by email. The backend user see the site as usual. All files are same for 2 domains (if they use same Joomla installation), but probably you can setup different paths in Media Manager settings and at least it will use different ...

Custom layout for usual Joomla's mod_login can be supplied and installed for free.

But you said "but we can supply a custom layout for usual Joomla's mod_login which shows the avatar from phpBB."