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