Can I Shoot Footage in Public without Consent?
Yes.
The freedom of photographers is often attributed to the first amendment right:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
Even though you're not officially a member of the press, the freedom of speech applies to any citizen, which does not only include the spoken word, but also texts and images.
There is a clause in the more recently passed legislation that leaves backdoors open for national security, which might be brought up by law enforcement when taking picture of airports, bridges, power plants and similar category buildings and structures, however this is not a blanket statement and unless you have ties to terrorist organisations you're probably fine.
Can I publish Photos taken of Strangers without Consent?
Yes, with exceptions. The displayed individual has a right to privacy which usually is strengthened if they're in a vulnerable situation or depicted doing something that would paint them in a false light.
You are allowed usually allowed to publish for your portfolio or within your capacity as an artist or for journalistic and editorial purposes, advertising and promitional material for companies is handled differently (see below).
Exceptions from publishing can be works also displaying artworks or spaces that require entrance fees.
Can I use Street Photos commercially?
Mostly. Model releases are recommended, however an important court case is the case of [Nussenzweig v DiCorcia] where the artist is granted the sale without paying damages to the subject, because artistic expression, even though the work is being sold, is not "advertising or trade" in the state of New York.
The New York law prohibits the use of a person's likeness, without consent, "for advertising or for purposes of trade." DiCorcia and Pace/MacGill argued that the photo represented "artistic expression", and was protected under the 1st Amendment and that the statute of limitations had expired for bringing a lawsuit.
Can anyone demand to see my camera or force me to delete my photo?
No. Photography can be prohibited in certain cases, but seizing equipment or demanding to see captured footage requires a warrant or while arresting you. Remember to ask if you're being arrested.
Also, in public, it is legal to record government officials carrying out their duties, if you're stopped by a security guard on private property, they can ask you to leave, you still don't need to hand over your equiptment or prove your innocence or good intentions.