Question:

Are there any classical sources saying that one can learn Torah and do Mitzvos to help others eg the sick?

Answer:

There are numerous sources for learning torah and doing mitzvos for a deceased person, the most famous one is saying kaddish for the deceased, see Meseches Kallah -2, regarding the famous story with R’ Akiva. Also see Zohar Chadash Acharei Mos 49b, Ohr Zarua 2-50,  Rema Y:D 376-4.  There is also sources regarding doing mitzvos that shuold be a zechus for the deceased person, such as learning mishnayos, see Aruch Hashulchan Y:D 376-13, see Chida Morah B’etzba 2-41. These things apply especially to children for their parents. Besides this, all mitzvos that children do gives a zechus to their deceased parent.

Regarding other people doing mitzvos for a sick person please see the folowing link where this was discussed. http://dinonline.org/2017/02/02/giving-merits-to-someone-else-for-healing/

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