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Admin Tools' .htaccess Maker is not fully compatible with multisites extensions. Everything else is.

While Akeeba Backup can backup multisite web sites it is NOT advisable. You can only backup the master site, including all its slave sites. This means that in case of a catastrophic failure in a single site you are obliged to roll back ALL SITES.

While multisites sound like a good idea you will quickly find out that the problems that come with a solution like this are much more severe than whatever problems you think you are solving.

Sorry, I think I got confused with some other multisite extensions.

Now that you explained to me how it works I better understand its description at the Joomla! Extensions Directory. Indeed, the extension you chose seems to simply redirect specific domains to specific menu items in the same Joomla! site. It is not a multisite solution, it's a clever trick which leverages core Joomla! features.

So, let me answer your questions again:

1. Is Admin Tools compatible with it? Yes, except the .htaccess Maker. The .htaccess Maker needs to know the domain name of your site to prevent redirection loops and simply because some of its features require the domain name to be hardcoded in the .htaccess File. Since your site will have multiple domain names this is no longer possible and you must not use .htaccess Maker.

2. Is Akeeba Backup compatible with it? Since you only have one site, yes, it is. As far as Akeeba Backup is concerned you have one Joomla! site. Yes, it listens to multiple domain names and serves a different page on each one of them, but it's one Joomla! installation. So yes, you can recover the entire site in a snap should there be a problem.

So, to make this clear: the extension you are using is not a multi-sites extension, it's a multi-domains extension. Joomla! already supports having multiple "sub-sites" in one installation. See the default English sample content in Joomla! 3. The extension you chose simply does a cool trick: instead of having to go to a specific menu item of the main site to see each sub-site it allows you to just use a domain name. Very neat trick, indeed. But you still have a single Joomla! site, which means that Akeeba Backup can still work with it.

> If I cannot backup the site at all, then there is no point of setting up the virtual domains.

As I said, you can backup the entire site. As far as Akeeba Backup is concerned you have a single site (which incidentally contains the data for every "sub-site" which is not really a site but a menu item and a trick to make it look as if it is a different site).

> I cannot find this on the internet. Is this the sample content comes with every fresh J3 installation?

Yes.

> If it is a core feature, will I still need this extension?

As I said:

The extension you chose simply does a cool trick: instead of having to go to a specific menu item of the main site to see each sub-site it allows you to just use a domain name.

If you want each domain name to direct to a different "site" then yes, you need that extension.

> So does this mean the virtual domains are like ghost sites?

Technically, they are different menus on the same site. There is no such thing as a "ghost site" :)

> Search engines wont be able to index them?

Of course they will be indexed.