السلام عليكم ورحمة الله وبركاته
Can a person do istighfar and other dhikr whilst eating and whilst using miswak?
جزاك اللهُ خيرًا

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.

The remembrance of Allah – among which is seeking His Forgiveness – is permissible at all times, and under all circumstances, except at times when it is confirmed that it is forbidden to mention Allah.

'Aa'ishah narrated: “The Prophet, used to mention Allah at all times (and under all circumstances.” [Al-Bukhari and Muslim]

Accordingly, there is no harm in mentioning Allah in the two cases you mentioned. However, the scholars mentioned among the etiquettes of mentioning Allah that the person should be at his best qualities, like ensuring that his mouth is clean (and smells good), if not then he should use the Siwaak (to give his mouth a good smell). Additionally, he should be facing the direction of the Qiblah, and ponder over what he says. This is undoubtedly the most complete manner.

Allah knows best.