Does changing a mission's difficulty affect engram drop rate?

Solution 1:

Mission rewards at the end are based on the level of the mission. Increasing the level/difficulty of the mission will increase the level of the reward.

You can only get items with light stat from missions that are level 20+, strike playlist, raids, and the crucible. They will not appear in regular strikes or missions under level 20.

You are correct about the more enemies meaning more chances for engram drops, but you could also just farm explore/patrol mode for engrams instead of missions.

I'm pretty certain the highest rarity non-level 20 items only go up to blue quality, and blue engrams drop consistently from level 1 to level 20.

Solution 2:

In my experience the drop rate does not seem to be connected to mission difficulty although it does seem to be connected to your level. The higher level you are the more stuff you get is how I found it although maybe its just luck.

I have gotten legendary stuff from farming earth patrols while none of the legendary or rare engrams that I have recieved doing level 24 missions strikes etc have given me anything better than rare. Factors that might increase the drop rate are headshots or major / ultra enemies.

Ive also heard that your cryptarch rank influences the outcome of engrams that you do find although it hasn't had much of an affect with only 1 of the 12 or so legendary engrams turning out to the legendary. His rank is only level 5 though and maybe its just RNG.