Are the acts of ablution divinely ordained, such that it is impermissible to deviate from them, or is it permissible to exercise ijtihad (independent reasoning) and analogy for washing a dirty hand after rinsing the mouth and nose, by analogy with washing the hands before inserting them into a vessel?

It does not harm to insert a hand, free from impurity, into a vessel of water for ablution (wudu') to scoop water; the basic principle is that the water does not change by that. However, if the hand is affected by blood or similar impurities that could impact the purity of the water, the hand should be washed before returning it to the vessel, in order to preserve the water's purity. This washing is not part of the prescribed acts of ablution.