F-Droid is another App store similar to Google's play store. The only different is that in F-Droid all the apps are Open Source and in many cases there are not available in the Google's play store. Like if you are looking to install VLC in Android, Google wont have it but you would find it in F-Droid.
There are no conflicts in having the clients for both app stores in the same device.
Android is Linux without the GNU layer on it, so Google came up with their own userland to replace the GNU stuff.
There are Operating Systems based on android that include some of the familiar linux tools for the command line. You could check on: CyanogenMod, PacMan, AOKP and more. To install them you have to unlock the boot loader on your device and boot from one the image of one of these. You can find instructions to do that on their site and in youtube ofcourse.
There are lots of steps to unlock the device and to flash another image in it. It is also really easy to brick your device trying to do that so I would suggest not to do it, at least for a while.
I am running Jelly Bean in my device and kitkat has been out for a while now. If I dont get and upgrade in the next few months I would probably try to flash a newer OS in my phone.
Enjoy your gift, and good luck to you.