I can't comment on just how far TC can be made to fulfull each item on your list, but perhaps I can make a few comments that will help you decide whether TC can be made to meet your requirements. You'll probably have picked up by now that the focus of TinyCore is to be *small* and *modular*. That means that it does not have gobs of automation already set up for various different kinds of installs and administrative scenarios. The standard unix-y tools are available to make your own, however.
TC supports several options for scripting. The simple 'ash' shell is included, the standard 'bash' shell is an optional extension, and there are several scripting languages beyond those in which you could write scripts to do the sort of automated installs you are after. Tinycore has some automated install-to-usb scripts which you can use as models. I don't think there is out of the box functionality to do just the kind of installs you are asking for.
Your final requirement is perhaps the thing that would take you the most work to create. It would require some kind of configuration server running on each local machine. TC does not do this by default. What it does have is an ssh server, so that you can get a shell on each machine remotely, and edit configuration files. So you can do a lot remotely, but for the most part it would be via a command line interface.
So you won't find that TinyCore is a turnkey solution for your particular case. If you need 'small and modular' and are willing to script your way towards the rest of the functionality you require, you will quite likely find willing help on this forum and the standard communications chanels for the scripting language you choose.
All the best.