Is a ransom (fidyah) obligatory for someone who performed Umrah on behalf of his deceased mother in Shawwal and wishes to perform Hajj in the same year, and is it permissible to perform Hajj on behalf of the deceased mother who has previously performed Hajj, knowing that the questioner has performed Hajj several times?

There is no harm in performing Umrah and Hajj on behalf of a deceased mother and dedicating their reward to her. If you are residing in Mecca or at a distance of qasr (the distance allowing for shortening prayers) from it, or if you are not from Mecca and performed Umrah then traveled a qasr distance and returned for Hajj as a mufrid (performing Hajj only, not combined with Umrah), then no damm (expiatory sacrifice) is required of you; either because you are considered among the residents of the Sacred Mosque, or due to your travel between Umrah and Hajj.