Does the Asus Nexus 7 allow me to create and install my own apps without using someone's app store (like Google Play or Amazon)? I want to create my own apps but not make them open for anyone else but me.
What you are asking is officially known in the Android Community as Sideloading.
The most "difficult" thing is to get a copy of the APK file of the app you wish to install, as you are bypassing the Google Play Store.
A very good comprehensive and informative article written recently on September 15th 2012 for Make Use Of entitled "How To Manually Install (“Side Load”) Apps On Your Android Device" by D.Stieben will give you additional insight into what you wish to achieve.