Hey, welcome to modding. Enjoy your stay
To answer your questions...
1) Yes. You should always run as admin, to avoid the many issues that can happen from not doing this.
2) When you open the geck and load Fallout.esm you then click Save. The first time you save it will ask for a name for your .esp file, or the mod itself.
3) Start by adding new interior spaces. Simply ones with short quests, and little NPC interaction or complicated scripting. First off, there is a tutorial. http://geck.gamesas.com/index.php/gamesas_Tutorial_GECK_Setup. It explains all the basics of the GECK and will really help you get started. Complete it, start to finish, before doing ANYTHING else. That website is also the wiki for the geck and contains helpful info on almost everything in the geck. Go there first with any questions as most of the time they will be answered. If not, then ask here.
4) Yes. Anything you change then save will show up ingame if you activate the .esp file you just saved.
5) If you have an account, on the website click Members and you will find buttons to make your own files where you can name it, upload things, and screenshots.
Have fun with modding. Just remember it takes time and patience to learn it, and make something thats good, but if you try it wont be so hard.
Gunmaster95