Salaah

Salaah becoming qadha

Q: This morning I was going to wake for fajar and heard the namaz and saw someone else in the bathroom so I closed my eyes and waited. I didn't realise that I fell asleep and when I got back up it was still dark so I rushed to the bathroom to make wudhu. I am a very doubtful person so making wudhu takes some time but I am trying to get better and did it as fast as I could. When I came back I looked outside and it was still a dark blue and there was little  yellow in the far back so I quickly started praying. I didn't think it was that late as fajar happens around 6. I thought it might be 6:30 at the latest but after I prayed 2 sunnats I checked my phone and it was almost 7. So I finished my namaaz. I know it is bad to delay prayer and I didn't mean to. I am so worried I might have prayed in the forbidden time but I remember reading that if you didn't mean to delay fajar prayer you can pray as soon as you remember or wake up so did I do the wrong thing? How do I fix this?

Reciting tasbeeh-e-Faatimi and Ayatul Kursi after Salaah

Q: Immediately after performing his Fardh and/or Sunnah Muakkadah Salaah, a person is unable to remain seated and recite tasbeeh-e-Faatimi, Ayatul Kursi and other wazaaif on account of some time constraints or traveling. Whilst he is walking to his car, or even whilst driving, can he thus recite Ayatul Kursi, tasbeeh-e-Faatimi and other wazaaif now? Or would it be better to delay it and recite it at another time when he can be completely dedicated towards it?