Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
I pray to Allah to bless you in your intention to perform Umrah and make it a rewarding experience for you and your family.
Now coming to the questions, let me answer them in their order:
1- There is only one tawaf for Umrah followed by Sa`i. It is called Tawaf. After the Tawaf, you pray two Rakahs near the maqam of Ibrahim facing the Kabah and drink from Zamzam water. Then, you should proceed to perform Sa`i, which involves pacing between the two hills of Safa and Marwa.
Once you have completed the seven rounds of Tawaf, you can free yourself from Ihram by cutting or shaving or removing a few hair locks.
2- You may also do it on behalf of your parents or beloved relatives, especially if they are not in a position to do so.
Since the Prophet (peace be upon him) allowed people to do so on behalf of their parents or relatives who were unable to do it, we are not justified in stopping people from doing the same. We may do well to remind ourselves that Islam is a religion of mercy; so let us leave it as it is.
Almighty Allah knows best.
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