Does Allah (swt) punish (in this life and in the Hereafter) the sons/daughters for misdeeds committed by the father/mother? Are sons/daughters rewarded (in this life and in the Hereafter) for the good deeds done by the parents?
Quran is very clear on this subject that good or bad deeds are not transferrable, as mentioned in
And no bearer of burdens will bear the burden of another. And if a heavily laden soul calls [another] to [carry some of] its load, nothing of it will be carried, even if he should be a close relative....
... And every soul earns not [blame] except against itself, and no bearer of burdens will bear the burden of another...
Whoever is guided is only guided for [the benefit of] his soul. And whoever errs only errs against it. And no bearer of burdens will bear the burden of another...
If you disbelieve - indeed, Allah is Free from need of you. And He does not approve for His servants disbelief. And if you are grateful, He approves it for you; and no bearer of burdens will bear the burden of another...
Although the answers by other brothers are actually correct, but let me complete them a bit:
Yes, Allah عز و جل punishes and rewards both in this life and in the Hereafter the children for deeds committed by their parents, and punishes and rewards the parents for the deeds done by their children, BUT only if those children or parents have any share, even if is slight and stealth to our eyes, in the good or bad deeds done by the parents or children, respectively.
Actually, this is a general rule, the deeds someone does have direct and indirect effects into his and others' lives: الأعمال ما تقدم and الأعمال ما تأخر are defined according to this obvious issue. Any one is responsible for every effect that his/her deeds might have, upon his own life or others' lives. This responsibility can grow grave when the persons tied together are parents and children, specifically when parents are highly responsible to treat their children and children CAN have great roles in the decisions their parents may make.
So we have many Islamic TRUE stories that a man was in torture in his grave but then relaxed soon after his son did a good job in his personal life, even without sending its rewards to his father, the reason for that is clear, the father had a share in that good job by treating his son correctly in that respect, now the son doing that is like the father in his own life is doing that. This should grab your attention to the important role a teacher or tutor can have, they will be responsible for many good or bad deeds of what their students do to the end of their lives, and even to the end of lives of all those who were affected by them in a sense or another:
أَلَمْ تَرَ كَيْفَ ضَرَبَ اللَّـهُ مَثَلًا كَلِمَةً طَيِّبَةً كَشَجَرَةٍ طَيِّبَةٍ أَصْلُهَا ثَابِتٌ وَفَرْعُهَا فِي السَّمَاءِ / تُؤْتِي أُكُلَهَا كُلَّ حِينٍ بِإِذْنِ رَبِّهَا ۗ وَيَضْرِبُ اللَّـهُ الْأَمْثَالَ لِلنَّاسِ لَعَلَّهُمْ يَتَذَكَّرُونَ / وَمَثَلُ كَلِمَةٍ خَبِيثَةٍ كَشَجَرَةٍ خَبِيثَةٍ اجْتُثَّتْ مِن فَوْقِ الْأَرْضِ مَا لَهَا مِن قَرَارٍ / يُثَبِّتُ اللَّـهُ الَّذِينَ آمَنُوا بِالْقَوْلِ الثَّابِتِ فِي الْحَيَاةِ الدُّنْيَا وَفِي الْآخِرَةِ ۖ وَيُضِلُّ اللَّـهُ الظَّالِمِينَ ۚ وَيَفْعَلُ اللَّـهُ مَا يَشَاءُ
Seest thou not how Allah sets forth a parable? - A goodly word like a goodly tree, whose root is firmly fixed, and its branches (reach) to the heavens,- of its Lord. So Allah sets forth parables for men, in order that they may receive admonition. / It brings forth its fruit at all times, by the leave of its Lord. So Allah sets forth parables for men, in order that they may receive admonition. / And the parable of an evil Word is that of an evil tree: It is torn up by the root from the surface of the earth: it has no stability. / Allah will establish in strength those who believe, with the word that stands firm, in this world and in the Hereafter; but Allah will leave, to stray, those who do wrong: Allah doeth what He willeth. [Al-Ibrahim, 24-27]
ِYou see, a dumb who is busy with his own life will be placed far higher in the Judgment day than a genius scientists who think has proved God not to exist(!) and make a vast number of lives go astray through a number of centuries ... Perhaps if scientists like Darwin, Dawkins, Hawkins and Freud (not judging about them, Allah knows better how to treat them, only examples for having significant effect in making people go astray) knew how they might make people go astray never have talked a single word.
I believe that parents are responsible for there children deeds because if your parents act bad you will act bad just like them. If your parents act good you will act good. Everything that you learn is from your parents. It is your parents responsibility to teach you good, we came to this world to worship the one and only God,allah(swt)
I think it's unfair to say that the parents are responsible for the way their children live after their passing. Every soul will pay a price for their own bad deeds. No parent can be blamed unless, in life, they wanted destruction amongst their children. Saying "my parents are turning in their grave" for what the children do, is, to me, not true!