Now, to many devout followers there penance is offered for the intentions of souls. However, the Church asks Catholics to do penance, but never actually tells them the reason. Do you believe it is a mortal sin not to do penance for souls? I am not talking about light penance such as fasting, but penance such mortification of the flesh and a day of silence.
No, certainly not. We are all expected to do penance of some sort, but we have a great deal of freedom in deciding what kinds of penance to do. (We are all, of course, expected to observe the days of fasting and abstinence from meat prescribed by the Church. But beyond that nothing specific is prescribed.)
I am not sure what kind of "mortification of the flesh" you are referring to, but we are not to take on anything unusual, harsh, or potentially dangerous to our health without the permission and supervision of a competent spiritual director.
Sometimes the best penances are the ones that come to us uninvited in the course of every day: taking criticism silently without retaliating; offering to to the dishes when we would rather take a nap; listening kindly to a boring person, passing up a second helping at dinner or a favorite dessert, etc. These penances are most pleasing to God because they come to us from His hand, rather than our choosing penances for ourselves (which can leave an opening for pride to enter in).
There are a couple of different kinds of penance.
We are called to do penance individually because of Original Sin.
God, the Father, promised to us a savior in Jesus Christ. Mary is a sort of bridge, a connection between Jesus and us sinful creatures.
To gain any merit our actions, prayers, offerings must be through Jesus Christ, for Jesus was the perfect sacrifice and is thus most pleasing to God.
Now, you can do penance in many different forms. Prayers, voluntary and involuntary mortification of the senses, offering up difficulties throughout the day, etc.
You can offer these for a variety of intentions for the living and deceased or even for yourself. For example, any partial and plenary indulgences can be applied to yourself or to another soul in Purgatory. We can offer up these for the intentions of others because we are all part of one body.
However, your question seems revolved around “intentions of souls” which i am guessing is the souls in Purgatory. We are not required to offer any sacrifice for the souls in Purgatory outside of Mass. In Mass there is usually some mention of the souls in Purgatory termed “We remember those who we have gone before us for whom we now pray” just before Consecration in the Novus Ordo (it varies depending on which text is used) or “Be mindful also, O Lord, of Thy servants and handmaids who have gone before us marked with the sign of faith, and who sleep the sleep of peace. To them, O Lord, and to all that rest in Christ, grant, we beseech Thee, a place of refreshment, light, and peace. Through the same Christ our Lord. Amen.” said in Latin in the Latin Mass shortly after Consecration.
So whether you realize it or not, when you are assisting at Mass, the sacrifice which you in union with the priest is offering up, is in part for the souls in Purgatory.