First, Welcome to the site.
There are lots of assets here, the routines thread will really amaze you at the number of ways guys pursue the goal of a really smooth dome.
But my answer to your questions:
#1) Should I put some cream after I shaved?
Definately, if only one I would recommend Nivea Extreme--it goes a long way, feels great and has a lite agreeable smell.
#2) Should I shave everyday ?
This depends, how fast your remaining hair grows and what your tolerance for stubble, aka velco. I have a relatively rapid growth and I can't stand stubble so I shave daily, generally after the gym. One thing to note, it helps to get a long lasting smooth dome if you can wait for an hour or so, like going to the gym, before shaving. Some guys only shave every other day bc that suits them. Remember, it's your dome, it's supposed to be comfortable. Do what you feel best doing.
#3) When I shaved both times with a bic razor, my head was "sticky" in a way and it was not comfortable to put a hat or something.
I wouldn't use any disposable blades on my scalp or face. My choice is the new M5 by Personna. Others love the Headblade, or the Mach III, or Fusion--I used that until I ran into M5. Also, a good shave creme is essential, I love my Headslick, but I've also used Baldguys Gel. I always use an "undercoat" of Shave Secret, too. If your scalp is oily or sticky, wipe it off with a Baldguys Headwipe, or in a pinch a wet paper towel in the bathroom where you work.
Welcome again, enjoy the freedom. All of us are here to answer your questions.