The decision to use screen protectors is different for each person. I used to have a screen protector but got tired of the way it felt. For me the glass screen is so much smoother without a protector. Screen protectors are good if you are in an environment where you might get sand on the screen which will definitely scratch it. I did not have that problem so I removed it. Since I was already using a case with a cover I figured the screen would be safe. After six months without a screen protector, my iPad screen still looks like new. I have not regretted removing the protector. Others would never consider going without a screen protector. I would say if you don't have a case, then a screen protector is probably wise.
I might also add that some screen protectors diminish screen clarity slightly, and many still result in fingerprints.
As to your iPad not charging when connected to iTunes, this is normal and is because the USB connection does not have the same power output as connecting your iPad to your charger. Even though the iPad says it is not charging when connected to you computer, in reality it is, just very slowly. Best bet is to charge it with your charger.
