"Now tell me does the freedom of speech allow Nazi content or racial comments."

Unfortunately, yes. These are ideas, and being offensive or incorrect shouldn't prohibit their expression. Many ideas once considered correct are now realized as misguided or wrong, but it is only through discussion that we come to these realizations. If discussing racism or Nazi Socialism favorably isn't protected, how we we talk about them critically? At one time people were punished for stating the Earth wasn't at the center of the universe.

"Should child pornography be allowed under freedom of speech or freedom of expression."

Child pornography is a different case, involving the physical exploitation of someone who is incapable of making an informed, mature decision to participate.

"Would freedom of speech allow me to question anybody's integrity without having proof of it."

If we use the terms 'slander' and 'libel' instead of the imprecise 'integrity' and 'mud slinging', then no. These differ from racist or hate speech, questioning intelligence, etc. in that you are referring to a specific individual, not a group in general. Ditto 'bullying'; this too is an action aimed at an individual.

"Freedom of speech is for presentation of ideas."

I agree, but who decides what ideas? Racism, socialism, religion, government policy; all are ideas.

"Yes you can criticize a drug pusher or human trafficker but isn't it different from belittling a mechanic or a city services worker or a dentist or an engineer and stereotyping them very negatively."

How is it different? What about someone who provides a beneficial service by selling marijuana illegally but exclusively to chemotherapy patients and glaucoma suffers, as opposed to an engineer who repeatedly designs structures that fail? Does your definition of free speech allow me to compliment the first and condemn the second?

For the sake of this discussion, I'll grant a 'right to anonymity'. Many who call for such forget that having such a right does not obligate others to provide an anonymous platform or otherwise facilitate it. A privately run web site, newspaper, broadcast station, or other privately operated forum may create it's own rules regarding anonymity, content, participants, etc. A 'right to anonymity' doesn't require private sites to permit anonymous behavior, only that you may exercise that right in your own publications without government interference.