Assalaamu alaykum.
Question 1: In a four Rak‘ah (unit of prayer) prayer, I read At-Tahiyaat and the Darud ("Allahumma salli ‘ala Muhammadin...") in the second rak‘ah in the sitting position forgetfully, now do I have to do Sujood Sahw (prostration of forgetfulness) or not?
Question 2: If I lost many prayers (by not performing them) before I started performing the prayer, then what should I do now? Should I start praying extra prayers to make up for these with the respective obligatory prayers? How do I compensate for the lost prayers? Kindly reply as soon as possible.
Please do not refer me to other answers. May Allah reward you.

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

You are not obliged to perform Sujood as-Sahw due to reciting the Ibraaheemic prayers (Darud) out of forgetfulness in the first Tashahhud. The scholars of Fiqh stated that if a person says a prescribed Thikr (expression of remembrance of Allah) in other than its place, his prayer is not invalidated and he is not obliged to perform Sujood as-Sahw, but some scholars held that it is desirable for him to do so. 

Ibn Qudaamah said:

“If one says Thikr at other than its proper place, such as reciting the Quran while bowing or prostrating, or saying the Tashahhud while standing in prayer, or saying the Abrahamic prayers in the first Tashahhud, or reciting a chapter of the Quran in the last two Rak‘ahs of the four-Rak‘ah prayer or the last Rak‘ah of the Maghrib prayer, and the like, then if he does so out of forgetfulness, is it prescribed for him to perform Sujood as-Sahw? There are two narrations in this regard: one of which is that it is not prescribed for him to perform Sujood as-Sahw, because the prayer is not invalidated by doing these things deliberately, so it is not prescribed to perform Sujood as-Sahw due to doing it out of forgetfulness, just like leaving the Sunnah acts of prayer (like placing the right hand upon the Left, etc).

The second narration is that it is prescribed for him to perform Sujood as-Sahw based on the saying of the Prophet ‘If one of you forgets, then let him perform two prostrations for forgetfulness while he is sitting.’ [Muslim] So if we say that it is prescribed for him to prostrate for forgetfulness, then this is only desirable and not obligatory, because it is an adjustment (a correction) for something that is not obligatory, so it is not obligatory in itself.”

As for the prayers which you had missed in the past, then you have to make up for them, even if they are many. Making up for the prayers is not restricted to making up for one prayer with every present obligatory prayer. Rather, you make up what is possible of prayers, whether few or many, every day. Some scholars specified this to making up the prayers of two days in every single day at least unless this affects your earnings (the time you spend in gaining your livelihood).

The commentary of Ad-Dusooqi on Ash-Sharh Al-Kabeer ‘ala Mukhtasar Khaleel (from the Maaliki School) reads, “The obligation is moderate: It is sufficient to make up the prayers of two days or more in one single day, as it is not enough to make up for the prayers of one single day only, unless one fears that he would not be able to provide for his dependents if he makes up for more than one single day in one day.”

For more benefit, please refer to fatwa 356791.

Allah knows best.