Must prayers that were performed without sniffing and expelling water (instinshaq and istinthar) or tayammum, and without complete bowing (ruku') and prostration (sujud), be repeated after a nose surgery, given that there was no initial medical consultation, but it was later found that these could have been performed?

First: Sniffing water into the nose (istanshaq) during ablution (wudu') is considered sunnah (recommended) by the majority of scholars, and obligatory by the Hanbalis. As for ghusl (ritual bath), it is obligatory according to the Hanafis and Hanbalis, and sunnah according to others. The prayer of one who did not sniff water into the nose is valid according to the opinion of the majority of scholars.

Second: It is not permissible to abandon prostration (sujud) and merely incline, except for one who is unable to perform it or for whom prostration would be harmful, based on the statement of a trustworthy doctor. Similarly, it is not permissible to omit the sufficient bowing (ruku') except for a valid excuse. Since prostration does not harm the questioner and he did not abandon prostration based on a doctor's advice, his prayer is invalid and must be repeated.