Installing Mavericks on a brand new machine that shipped with Yosemite pre-installed?

It's a great big "Maybe".

The installer will likely decline, but it will really depend on the hardware and drivers the hardware needs.

Here's what you do:

Get an external drive. USB > 8Gb will work. Plug this drive into a Mavericks-compatible machine and install Mavericks on the external.

Now plug it into the new Mini. Reboot holding the Option key, wait for the boot picker to appear. Choose the external with Mavericks.

If you get a kernel panic, the new Mini will not run Mavericks.

If it boots to the desktop, you're good. Get some cloning software like Carbon Copy Cloner, erase the Mini's internal drive and clone Mavericks from the external drive to the Mini.

No, not possible. It can be possible when the a mac was delivered before with Mavericks. But the 7,1 was delivered with Yosemite from the beginning.