Yes. The disc will need to be formatted before you can install Mac OS X. This is probably not the question to which you wanted an answer.
This will depend on the specs of the Mac mini in question but in general, yes.
if you formatted it no, you would have to reinstall the old OS again
It depends on the age of your Mini. New models run Yosemite and will run newer OS's. You might download MacTracker. It's an application that will tell you most everything you need to know about every Mac made.
There are apps that will only work with Mac OS X which can be on any Mac computer - iMac, MacBook, Mac Mini etc.
If the hard drives are formatted for the Mac, Mac OS Extended (Journaled) with a GUID partition table then yes.So more then likely NO, unless it specifically states it in the manual.
It is illegal to run Mac OS X on anything but Apple computers.
Yes, when you "reformat" it erases the hard drive completely. It will be an empty drive, you will have to reinstall the OS from your disk/Apple internet or reinstall from a back-up after you do it.
Providing a Mac mini is running Mac OS X 10.5 (Leopard) it can be used to make iPhone apps. Apple provides separate versions of the iPhone Software Development Kit for Leopard and Snow Leopard. The SDK requires the Xcode development tools which are included with the Mac OS X installation disc.
The main OS for Mac can be found on a Mac when it is switched on. The current main OS is called Mac OS X.
The hard drive will need to be formatted (rather than completely blank) and then Tiger can be installed on it.
The Mac OS X Server software (See links below) can be installed on any modern Apple Mac computer or can be purchased ready installed on a Mac Mini ready to go.
I have a Mac OS X but I know Mac OS is a computer made by the brand/company, Apple.