How to get a mission to continue the story

I've definitely seen kills while playing alone from nearby players help me: I've seen them progress me towards killcounts, and I've seen them cause the generation of things I need to pick up.

Yes this essentially adds a damager modifier of 3x (sorry I couldn't find a Bungie source to quote for that number) for both damage given and received.

This reddit thread contains some experimental numbers are sums things up quite nicely:

Now this makes a lot of sense. It seems that Bungie applies the 300% modifier FIRST and then the elemental shield reduces that number by 200% due to the non-corresponding elemental damage. This would make the modifier 150% for the burn element to the non-corresponding shield element.

In conclusion, it would be smart to assess the weekly nightfall to see what shield types there are, then select weapons accordingly to take down enemies the fastest. For example, if most of the enemies have void shield, yet the burn element is arc, take a void weapon to take down the shields then use the burn element to finish them off. This is smartest in a scenario where you're taking down an ultra or heavy mob.

So even with weapons matched to the appropriate burn modifier it may still be better to have a variety of damage types especially if running a mission where there are a mixture of enemies with different shields. For example when running hive missions wizards have solar shields but knights have arc shields so having both solar and arc damage can be helpful.

Of course you can just take the appropriate matching damage type and hope you find lots of ammo for your weapons. This is where weapons with regenerative ammo/extra ammo pickup perks can be exceptionally useful if they match the burn modifier e.g. Invective or Ice Breaker for solar damage to name a couple of examples.