Is there a word (probably a verb to describe the action) for the below situation:
When someone throws a sarcastic remark to you, especially in front of other people, not in a playful manner but rather an offensive one, meaning to dampen you.
It could be more of a slang term, an "urbandictionary-word", than a conventional English word.
There will be a lot of such words and expressions (so I hope you'll resist the urge to select an answer prematurely). The one I'll propose is cut down. Here are a couple sample usages:
I didn't appreciate the way he cut me down during the meeting.
Maybe he cut down Susan during the meeting, but I don't think he did himself any favors.
The phrase is a shortened form of cut someone down to size, meaning "to put someone in their place."
It should be noted that the phrase cut down has several meanings, so you'll want to provide ample context to make it clear what you are talking about.
If you wanted to use a noun rather than a verb, you could try slap in the face. It is slang for "an insult."
Bob's remarks were a real slap in the face.
By the way, the word real in that example sentence means "serious," not "actual" (i.e., Definition #5, not Definition #1a, in this dictionary). Contrary to how it might be interpreted, that sentence doesn't mean that Bob was literally slapped in the face.
I think the closest you will get is with "sardonic".
Someone with a "biting" personality or who is prone to making "biting" remarks is making the kind of offensive, rude, and deragatory insults of which you speak. This can be true both in sarcastic and straight-forward situations, although typically biting satire is sarcastic.
The verb snark may be appropriate. It means “To express oneself in a snarky fashion”, where snarky means “Snide and sarcastic; usually out of irritation, often humorously”. Example: “He snarked at her all through the meeting.”
Also consider verb backbite, “To make spiteful slanderous or defamatory statements about someone” or “To attack from behind or when out of earshot with spiteful or defamatory remarks”.
The phrase “do him down” or “do her down” means to denigrate someone by making critical remarks intended to damage the person's standing or reputation.
The way I would express this is:
He's always putting me down.
This is similar to "cut (someone) down" and is an informal way to describe someone belittling you. If it's more important to indicate a snarky comment that you didn't appreciate, you might instead say:
He made a joke at my expense.
To my knowledge, there is not a term that is specific to overly hurtful sarcasm, but if you trust context and tone to carry your meaning, you could also say:
He's always messing with me.
You may use the verbs 'calumniate','besmirch', 'asperse' or 'slander'.
She slandered/calumniated/aspersed/besmirched me in front of everyone that day
Surely the best way would be: "S/he was very sarcastic about me".
In [Lowland] Scottish, it is often abbreviated to "sarky", e.g. "S/he was very sarky to me".
The Chambers Thesaurus entry for sarcastic (link) is
adj
ironical, satirical, mocking, snide, taunting, sneering, derisive, derisory, scornful, sardonic, jeering, scoffing, scathing, cynical, incisive, cutting, biting, caustic formal disparaging, acrimonious, acerbic, mordant colloq. sarky
You can take your pick from there, pretty much.
S/he mocked me,
s/he taunted me,
s/he sneered at me,
s/he derided me,
s/he jeered at me,
s/he made a scathing comment about me,
s/he made a really cutting (or biting) remark about me,
s/he made a really caustic comment about me
