Is it legislated to raise one's hands in supplication in the mentioned situations, or is that an innovation?

The default is the permissibility of raising hands in supplication (Du'a), except in instances where it has not been authentically reported that the Prophet, peace be upon him, raised his hands, such as supplicating in prostration (Sujud), between the two prostrations, and after the final Tashahhud.

A person supplicating should raise their hands outside of prayer. As for the Imam raising his hands during the Friday sermon, it is disliked except in the prayer for rain (Istisqa'). It is also not legislated for the congregants to raise their hands in 따라 to the Imam during the Friday sermon, except also in Istisqa'.

It is permissible to raise hands when supplicating between the Adhan and the Iqama, and after concluding a lesson or a gathering. As for supplicating between the two sermons, some scholars hold the view that raising hands during it is permissible, while others consider raising them an innovation (Bid'ah).

The preponderant opinion is that raising hands after prayer is permissible, even though some verifying scholars disagreed with this due to the absence of a textual proof for it.