2).She takes after her dad

3).I took my brother along to the kindergarten

4).He spent two days taking his car apart.

5).In difficult situations he can’t shut up. He is always take taking his opponent apart.

6). It was a difficult question, so I decided to take him aside to another room.

7). My mom has taken me away to the home. (This one is grammatically correct, sort of - if your mom has taken you to your house, then I would say My mom has taken me home.) 

8). I have to take back all those (maybe "of the" would be better) books which (I would never hear or see "which" used here in my region, but it's technically correct) I took out from the library.

9). She had to take down all of the words what that he said.

I had never realised how complicated the verb "to take" is before. It has so many meanings. 

2). She takes after her dad. - Correct but remember to end sentences with full stops/periods.

3). I took my brother along to the kindergarten. - Correct, but I would just say 'I took my brother to the kindergarten'.

4). He spent two days taking his car apart. - Correct

5). In difficult situations he can’t shut up. He is always tearing his opponent apart. - 'To tear' is a better verb here than to take.

6). It was a difficult question, so I decided to take him aside to another room. - I would say 'It was a difficult question, so I decided to pull him aside into another room' or 'It was a difficult question, so I decided to take him into another room'.

7). My mom has taken me away to the home. - My mom took me away to the home. Note that you have used the American spelling of 'mom' rather than the British 'mum'. This is not an issue, just something to be aware of.

8). I have to take back all those books which I took from the library. - I have to take back all those books which I got at/from the library. Taking would imply that you stole the books from the library but it would only lightly imply that, it is not a verb that would normally be used in the context of taking something with permission. The 'take back' use is correct, though.

9). She had to take down all words what he said. - She had to take down all (of the) words that he said. 'Of the' is optional, but it sounds better with it. 'Take down' can also be used in a combative context for example a professional fighter may say 'I am going to take down my opponent in the match'.

An additional note, I would generally stick with either '10)' or '10.' in an ordered list because because '10).' is too much punctuation. Leave a space before the beginning of the sentence. I personally prefer the number followed by the full stop/period, '10.'.