The prepositions of and about add some indirection to the sentence. For example:
I know him
I know about him
I don't know about him
I know of him
The first means that you know him personally: the second means that you have information or experience of him. The third sentence could either mean that you don't have information about him, or that you do have information and it's not very favourable. The final sentence means that you have heard about him (you know that he exists) but you don't know him personally and don't have any information about him.
Looking at the sentences 1 and 2:
1) I don't know about how you took care of her.
2) I don't know how you took care of her.
The first sentence suggests that you don't have any information about how you took care of her: this might be used to indicate that the speaker doesn't know whether you did the job well or not.
In the second sentence, how is ambiguous: it could relate to what care was given (what specifically was done), or to how it was possible for you to give care to her (what obstacles were overcome).
The latter might be the case if both speaker and listener know that the person whom the listener took care of has in the past been unkind to the listener. For example, if she is the listener's mother, and she had an argument with the listener five years ago and hasn't spoken to her since.
For sentences 3 and 4, good
is often used as an adverb in American English, but in Brtish English, we think that good
is an adjective and well
is the corresponding adverb. For avoidance of doubt (as the lawyers say), I have replaced good
. The preposition in
makes sense if party
is a political party. If you want to talk about a social event, you would have to use the preposition at
3) I am thinking of how well he will perform in the party.
4) I am thinking how well he will perform in the party.
Sentence 3 indicates that the speaker is in the process of assembling information about how well he will perform, and has not yet made their mind up.
In sentence 4, the speaker is stating an opinion, and almost-fact, about
how well he will perform.