1) Yes. You can do that in 2 ways:
-By Selecting an Empty Save Slot after loading the game (or after selecting "Reload" in the main menu). You can also delete an existing Save File by Selecting said Save Slot and pressing the Key D.
-By selecting an advanced Save File, and pressing the Key 1. This will start a New Game + where you start the Free Mode anew with some perks.
2) Yes. After taming the same species of Fapet 5 times, you can go to "TC" and ask her to shave the fapet that is following you. (Must be from the same species you captured 5 times previously)
3) As long as the clothing is unlocked and is selectable in a Save File from the V1.4 Hotfix, yes.
4) Those are Shinny Bunnies you can find in the "Winter Zone" as part of the Christmas Event (Free Mode Exclusive)
5) There are 2 stages:
+10 Captures = Unlock Shirt and Pants
+20 Captures = Unlocks Hairstyle (females only) and accessories