Apparently what those of you who support gun ownership believe is that the government is not able to adequately protect you, therefore you need guns to supplement their effort.
It goes a little deeper than that. I don't believe I should have to depend on the government alone to protect me from the run of the mill criminal. The cops wouldn't have stopped my friend's intruder in time, but a bullet from her gun would have, if she had one. Everyone has a fundamental right to protect themselves. In the article I linked to, the attitude the British took was that people should leave their protection up to professionals. That is only workable if the "professionals" can be everywhere at once. They can't. There weren't any cops to assist the students at that school in Virginia, but the students assisted themselves and reduced the body count.
The "leave it to the professionals" ties in with your second question: no, we shouldn't do away with the police. They have the manpower, the forensic teams, et cetera to track down criminals. If someone is in a hit and run, they shouldn't have to track down the plates and analyze the tire tracks for themselves. The average person can't or won't analyze paint samples. The average person lacks the authority to interrogate suspects and witnesses. The police do more than just protect; they'll always have a function to fulfill regardless of whether a populace is armed.
If everyone is armed, will the criminal be deterred? That seems likely--when have you ever heard of crimes at the Dunkin Donuts? Or whichever donut store the cops prefer Also, Switzerland requires every household to have a gun. They also have very low gun crime, in fact little violent crime at all. Coincidence?