Is it permissible for a person who broke fasting in Ramadhaan because of being sick to pay a ransom?
If there is hope for him/her to be cured, then he would be required to make up for the missed fasting days when he/she recovers, and no ransom is due on him/her. If his/her disease is incurable, then he/she would be required to pay a ransom for each missed fasting day, which is (600) grams of wheat, or rice.
Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram(Marriageable) to her.
Is it permissible for a man to propose to a woman during her `Iddah(waiting period)?
It is impermissible to propose to a woman in her `Iddah, but there is no harm in making an implication to such an end in case she was observing it due to the death of her husband.
What should a sick person, who is incapable of fasting, and who didn`t make up for missed fasting days, do?
One who has missed fasting days is obliged to make up for them, but if he/she wasn`t able to because of an incurable disease , or old age, then he/she has to pay a ransom which is feeding a needy person for each of the missed fasting days.