Is it obligatory to repeat prayers that were performed while continuously passing gas, feeling it escape during ablution and prayer, and while praying seated with gestures for bowing and prostrating?

If the emission of gas is frequent, then either there will be a time when it stops, in which case you should perform ablution and pray during that time, or its emission continues without interruption. In the latter case, your ruling is that of someone afflicted with سلس (incontinence). You should perform ablution after the time for prayer begins and then pray, and whatever is emitted from you will not harm your prayer.

If you are afflicted with سلس while standing but not while sitting, then pray sitting. However, you must prostrate on the ground and not merely nod. If the gas emission stops while you are sitting but not while prostrating, then you are required to prostrate on the ground.

It is recommended to repeat the prayers you performed sitting, with nodding for bowing and prostration.