The following example would also sound clumsy in English: There is the possibility to find /of finding a solution. I have never seen the infinitive after possibility and that's why I would mark it as a mistake. The main problem with above construction is, however, that both the infinitive and the gerund sound clumsy. English people would prefer: It may be possible to find a solution.
'to reach an aim' is also pure German English, and I would mark it wrong. It should be: 'to achieve an aim'.

He went to a shop for buying food  is wrong. Here the infinitive must be used. 'for buying' suggests a reason (weil er Lebensmittel kaufen wollte). 'wollen' is not expressed in that 'for buying' - construction.

'people's life / lives'
'we live our own life / lives'
'we made up our mind / minds' 
Above three examples are possible, but it is more common to use the plural forms.

The same applies to the so-called Saxon genitive and the of-genitive. If one of the two nouns is a person I would prefer the Saxon genitive. But there are, of course exceptions like 'a two months' holiday' (with a space of time). As there is no convincing rule about the two genitives I would mark neither of them wrong. It is a matter of clumsiness or not. -


The English teacher's nightmare
Spot the 10 mistakes:

This music- box is playing since four hours. Much longer than it seems desirable. There must something happen to stop it; the public opinion will not tolerate it! If only these people would be prepared to renounce their pop music for five minutes!  I have not come here four hours ago to hear such music.! Therefore I go at once. -
The correct version:

This juke-box has been playing for four hours. Much longer than seems desirable.Something must be done to stop it; public opinion will not stand for (tolerate) it! If only these people would be prepared to do/go without (give up) their pop music for five minutes. I did not come here four hours ago to hear that kind of music! So I'll go at once. -