how to use "care about" and "care" as verbs

Question

I'm learning English. I have some questions about how to use "care".
If you use "care" as a verb, you usually need "about". But in some cases, you don't have to use "about".

1. I didn't care about what anybody thought.
2. I didn't care what anybody thought.

My dictionary says both are acceptable, when the object is "what" or "why" or other words start with "wh" including "how" and maybe "if".
Then I have three questions.

Are the two sentences considered to be correct?
Can you remove other prepositions after other verbs when the object starts with "wh"?
Can you use "care" without "about" when the object is just a noun?

Answer 1

Are the two sentences considered to be correct?

Both are grammatically valid. I think sentence 2 is better writing. I find the use of "care about" in sentence 1 to be awkward. But tht is a judgement question, ther is no rule in the matter.

Can you remove other prepositions after other verbs when the object starts with "wh"?

That is too broad for a single answer. In some cases a preposition is not needed, in other cases it is. It depends on the verb, and on the way in which the verb is being used.

Can you use "care" without "about" when the object is just a noun?

"I don't care {noun}." is not a valid construction. But other prepositions can be used, "about" is not the only choice. If "care" is followed by an adverb, not a noun or a noun phrase, no preposition is needed.

Examples:

• I don't care for Italian food.
• I don't care for Laura.
• I don't care that you dislike me.
• I don't care easily, but when I care I care strongly.

Answer 2

1. Are the two sentences considered to be correct?

Yes.

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2. Can you remove other prepositions after other verbs when the object starts with "wh"?

Yes, although it is not common. Here is one example, using the same preposition:

I often wonder about what I'll do.
I often wonder what I'll do.

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3. Can you use "care" without "about" when the object is just a noun?

Generally, no. M-W gives only two senses of "care" as a transitive verb. The first (with the example "doesn't care a damn") includes a noun phrase, but I would consider that noun phrase an adverbial phrase rather than an object. The second ("if you care to go") only includes an infinitive object.