Hello Lucy, 

I hope I can help you with this correction and my simple explanations :)

The correct sentence would be:

Nach unserem fantastischen Urlaub habe ich von meiner Familie Abschied genommen.

It may be grammatically right but we wouldn't say that in Germany.

If you say it that way it sounds like you or your Familie is about to die or your wont see each other for a long time.

Nach unserem fantastischen Urlaub habe ich mich von meiner Familie verabschiedet.

That's the way you would say it here in Germany. The "Abschied nehmen" is replaced by "sich verabschieden". 

It's like the good old good-bye in english.

2) Anstoss nehmen
Ich bin losgegangen, weil ich Anstoss genommen hatte.

To be honest, I don't really understand this sentence. The meaning of "Anstoss" here in Germany is equal to the English "kick-off" and is football-slang. Of course there is also a other meaning here but we don't really use it in that way. Atleast I don't.

A correct sentence with "Anstoss" would be:

Das ganze Stadion erwartet den Antoss.

3) Einfluss nehmen
Ein Vater nimmt immer viel Einfluss mit seinen Kindern.

I'm going to try explain this sentence with the english "to have impact on someone".

Ein    Vater     hat     immer     viel      Einschluss     auf      seine    Kinder.

A      Father    has     always    alot     impact            on       his        children.

I know the english sentence is wrong but it's just to show u what I mean.

4) in Kauf nehmen
Wir wussten nicht die Nachtricht glauben, aber nach einige Tagen haben wir sie in Kauf genommen.

Again I need to guess the meaning of this sentence and try to fix and translate it.

Wir haben der Nachricht nicht geglaubt, aber nach einigen Tagen haben wir sie in Kauf genommen.

Translated:

We didn't belive the message but after several days we embrace the possibility of losing.

In Germany we use "in Kauf nehmen" almost only for things with a possibility of losing.

For example it's a cloudy day with  dark coulds. Your significant other tells you it's going to rain and that you're going to get wet.

Than you'd answer "Ich nehm es in Kauf" or "Nehm ich in Kauf". 

Use it when you take the risk of something bad happening if you do something but in your opinion its worth doing it because

the consiquences aren't that worse.

5) Rücksicht nehmen
Obwohl wir uns oft streiten , nehmen wir uns noch Rücksicht.

The meaning is right but the grammatic is a bit wrong.

Obwohl wir uns oft streiten, nehmen wir Rücksicht aufeinander.

The "aufeinander" states what persons are meant. You could replace the "aufeinander" with "auf" and a person.

"auf dich", "auf unsere Eltern", "auf andere" oder "auf unsere Mitmenschen".

Questions? Write me!