Is stolen money to be returned at its original value or its current value, and to whom should the money be given if the grandmother is incapacitated, and does caring for the grandmother mitigate the sin of theft?

You must return the stolen money in dollars, regardless of its value at the time of the theft or now. The money should be given to your grandmother if she is of sound mind, so she can spend it on her needs. If she is not of sound mind, the money should be given to whoever is responsible for her affairs. As for your care for your grandmother, it does not alleviate your sin. However, if the theft occurred when you were young, before the age of accountability, then there is no sin upon you, according to the hadith: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he grows up, and from the insane person until he becomes sane or recovers."