Title pretty much says it. I've unsuccesfully searched through Google and could not find any info.
It definitely looks like the pace at which you kill an enemy (be it a mob, elite or a boss) affects the quality of the loot or the likelihood of getting a legendary drop.
Is that the case?
Or perhaps a different effect, like cummulatively having killed a certain amount of enemies within say a couple of seconds, leading to a higher legendary drop chance?
Fortuitous Amulet of Luck
Remember those? That was back in Diablo 2, when Magic Find was a stat you could view on your character. What number was best? I remember liking 327%, but even some lower percents seemed to work well too.
Since then, Blizzard has become more guarded about Magic Find and the secret recipe behind the scenes. Largely, this system is guessed at, and since the Real Money Auction House of Diablo 3, it has become more industry standardized for random "payouts".
Randomization in slot machines
In order to analyze this, it is important to look at what that industry standard is, and for that you can look at slot machines. Nowhere in the world is random more orchestrated than in a slot machine, which are mandated by laws in most areas to pay out a certain percentage of their intake (generally it must pay out 95%).
If the machine doesn't pay out a proper amount, it may become in the red and need to pay out faster in order to balance the deficit; the so-called hot machine effect.
Diablo uses loot tables
Diablo seems very similar, whereby there will be cold and hot spells of drops. I mention all of this, because the point I am making is that the largest factor here is going to be the amount of tries at a roll. The roll itself for loot tables can be adjusted behind the scenes based on certain criteria, and in order to make sure that you are playing in the balance, it is important to make sure that you consider cold and hot spells are in part by design.
While certain events require finishing within a timer, that is generally displayed. The prime factor to focus on here is going to just be the breadth of attempts at rolling, and the type of loot table you are rolling through.
Does killing enemies quickly increase likelihood of better drops e.g. legendaries and better legendaries?
Not from what I have seen, and it has also not been the case in previous versions of Diablo. The loot table is the greatest factor and it is static.
Or perhaps a different effect, like cummulatively having killed a certain amount of enemies within say a couple of seconds, leading to a higher legendary drop chance?
Not necessarily. Normal mobs have a loot table coinciding to global drops, and that legendary percent is very low. However, the game does attempt to coincide the rates with what happened to some degree. If you have a 4% drop rate for legendaries on the current mob, and you kill 300 of them, you may see that you get 2 drops in a short time frame as the game attempts to correct.
This scenario though is highly unlikely, just as getting 26 of the same color in a row in roulette is (#montecarlo); however, it does happen and it can lead to perhaps unusual effects.
Your best bet is to focus on finding a mob or drop with a very good loot table and balancing that against how often you can get a roll on the loot table. For example, world bosses guarantee a unique drop on their first kill, but are difficult to farm consistently following that and their loot table changes for repeat kills. None of the items you find prior to level 100 are going to have much meaning though, so it is also important to consider balancing experience gain or unlocking progression with your farming.
There are a few factors that decide what type of loot you can find:
Other than that there are no known limitations or benefits of loot tables between enemies, or logic that gives you better chances at better gear, such as kill speed. All we can assume for now is that the drops are random, with the above rules as limitations.
