Is it permissible to supplicate with other prayers after the well-known Qunut supplication and before prostration? And is shyness from being seen by others during supplication and prayer considered a sin?

There is no legal impediment to adding more invocations to the Qunut. It is preferable to recite the supplications that have been transmitted in the Sunnah. As for abandoning obedience to Allah out of shyness from people, this is a blameworthy shyness in Islamic law. Whoever is afflicted with this should strive against their soul, repel ostentation, and perfect sincerity. For Satan invites either to abandoning good deeds or to ostentation, and abandoning a good deed for fear of ostentation is ostentation itself.