Sitting for tashahhud in the first rakaat of Zuhr Salaah and making sajdah sahw

1. I was reading 4 rakaats fardh of Zuhr Salaah. After the first rakaat, I sat down and started reading tashahhud. I realised my error, but I completed the tashahhud, stood up and after the second rakaat did the tashahhud again and completed the rest of my salaah as normal. I did a sajdah sahw at the end, but I didn't recite the tasbeeh in the sajdahs. Is my salaah valid?

2. Previously, I was also reading 4 rakaat fardh of another salaah. In the second rakaat, after reading Surah Fatiha, I untied my hands to go in ruku. I quickly realised my mistake, tied my hands again and read a surah and continued the rest of the salaah as normal. Was I meant to read sajdah sahw in this situation?