To me the flag seems to have a feuille morte field and maybe a very dark (blue?) field?
It almost certainly is the Orléans-Infanterie
drapeau d'ordonnance which had blue and
feuille morte quarters (1-4 and 2-3 respectively). This is consistent with the soldiers' uniforms and the musicians' livery coats.
Did there exist any Orleans hussars?
No, but the Colonel-Général hussar regiment was created in 1783 and given to Louis-Philippe d'Orléans, Duke of Chartres, who was appointed Colonel-general of the hussar mounted arm. He is almost certainly the figure depicted here in the Colonel-Général uniform uniform of a dark blue dolman and black mirliton with red flame. His breeches should be blue instead of red, but that may be an error on the part of the painter. Here's a famous portrait of Louis-Philippe d'Orléans by Reynolds :

On the death of his father in 1785, Louis-Philippe inherited the title of Duke of Orléans and was no longer referred to as the Duc de Chartres. He later sided with the radical Montagnards during the French revolution, changed his name to Philippe-Egalité and voted in favour of the execution of his cousin Louis XVI. He then fell out with the Jacobins and was guillotined in 1793, essentially for being born a Bourbon.
The shabraques of the princely regiments seem to should have been red, but the officer - if he is from the cavary regiment - has a dark blue shabraque?
They weren't necessarily red by the 1780's :
Orléans-Cavalerie is recorded as having dark blue schabraques by the 1780's, and the two mounted commanders next to the Duc de Chartes are obviously senior officers from that regiment since their blue coats have light blue facings.