'take a look' and 'look'.
Although there is not a big difference, I think we can categorize them according to politeness.
What I mean is that when people say take a look at the something, they are more like ordering. On the other hand, if I give an example, when people tell somebody to look/have a look at an article from a newspaper that might get interest of him/her, it sounds like offering.
However if you ask my opinion, they are both equivalent in meaning. You can use both in your daily life without hesitation ; when you go to a bar you you could say take a drink or have a drink or just drink and likewise, have a shower and take a shower. I don't see a clear difference between them.
For your second question I can say that knowing something/somebody describes result,but getting to know implies more like the process that you are meeting somebody and learning his/her personalities. İf you know Tom, that means you know Tom :) ; you know he is after Jerry, and he will never give up catching the Jerry. On the other hand, If you have just started to watch Tom and Jerry, you will get to know what is going on in the animation in time.
While trying to answer this question, to be honest I really struggled :) . When people ask questions here, I can answer most of them without difficulty. However, your Q was quite hard. I hope I helped you to have a clearer idea .