Could you please check these sentences, too? I tried to rephrase the first lines again.
1) Hamlet wonders whether it is better (for him) to endure the assaults of shameless fortune ("the slings and arrows of an outrageous fortune") or to fight against his mass of troubles.
2) He is doubtful whether to surrender (give in) to fate or to fight against it. /He is doubtful whether he sould surrender to fate. ...
2) Human beings are born (or bound?) to suffer.
3)He asks Hamlet to avenge him (also: to take revenge/vengeance on Claudius for his own murder /are they possible???) but to leave his mother’s punishment to heaven.
1. I'd leave out "shameless" but the rest is fine.
2. OK (watch the spelling of "should" however)
3. Yes, the variations are possible, but wordy. Your first phrasing is the best.
1) The sentence seems to be rephrasing Hamlet's dilemma of whether he should passively endure the challenges and misfortunes thrown at him by fate ("the slings and arrows of an outrageous fortune") or actively confront and combat his troubles.
2) The second sentence is attempting to express the same idea as the previous one. Some possible rephrasings could be: "He is uncertain about whether he should succumb to the forces of destiny or resist and fight against them." or "He is unsure whether he should yield to fate or oppose it."
3) This sentence appears to convey the notion that human beings are destined or fated to experience suffering throughout their lives.
4) The last sentence states that someone (presumably a character in Hamlet) implores Hamlet to seek revenge against Claudius for their own murder. As for the phrasing, "to avenge" and "to take revenge/vengeance" are all valid alternatives. Regarding the mother's punishment, it suggests that it should be left in the hands of divine judgment or the workings of fate.