I am currently running 6 copies of Joomla 3 in a single hosting account and would love to make it just 1 copy!

I have 3 sites which are live sites with no sharing of users (except for me as administrator) or content. There is also a test version of each of the live sites,

Can MightySites help me ...

The run on the same Hosting Package.
How does the connection between the Forum and the Joomla Page (should) work?

Here is an overview how it will be:

... have usual media folder.
Second site (slave) should have a symbolic link (symlink) named as this folder and pointed to folder of master, so actually both sites will use same folder and same files.

Symbolic link is a kind of Shortcut in Windows.

You can ask your hosting support to create a symlink.

There is no demo because it's useless without access to our hosting panel and you just won't see anything in frontend - these will be usual Joomla sites.

It's possible to share VM database data, so you will have same VM on all sites. VM doesn't have multi-store functionality, better check MijoShop.

Yes, exactly, MightySites allows to run multiple Joomla websites from single installation of Joomla on master site.

Slave sites are created as parked domains in hosting panel - so they use same files.
But each site can use own database tables and hence have own contents.

... is auto-created to master.com root server folder, so both http://master.com and http://master.com/xxx are resolved from same server folder and hence use same files.

Slave sites like otherdomain.com should be first created in your hosting panel and pointed to same server folder as master site uses.

Yes, it's possible to have slave sites like http://domain.com
Such domain name can also be created as parked domain (domain which shares server folder with master site and hence use same files).
Please ask your hosting provider about such option first.

Copy files and database into separate hosting account, update Joomla config and uninstall MightySites.

... delete any file all sites are affected.

Sharing files is required of you share database data, otherwise i.e. user will upload avatar or gallery image in site A and it will be missed in site B.

You don't need to purchase xxx.master.com, such domain can be configured right in your hosting panel.

Yes, you can create a virtual site like xxx.mainsite.com (this domain first should be created as parked domain in your hosting panel) or mainsite.com/xxx (no need to create domain).

There is an ability to clone database tables of current site for this new site.

Next you can make changes in this site ...