Is it permissible for my father to pay for my Hajj expenses so that I can be a mahram for my sister, given that I am currently unable to afford its costs, and would this be counted as a Hajj for me even though it's not from my own money?

It is permissible to perform Hajj using funds provided by one's father, relatives, or friends, and it is not a condition that the money be from one's own private funds. However, performing Hajj with one's own money is better, so that one may combine the reward of Hajj with the reward of the expenditure.