Simply dividing the number of successful drops by the number of drops will give you a proportion. For example, 20 successful drops out of 100 drops is a 20% chance of getting the successful drop. This is the sample proportion.

You are correct that you will have to do it over and over, as any statistics you calculate are based on the sample, and you cannot have 100% confidence in them. You can, however, repeat it enough times to get a confidence level of 95%, 99%, etc. Also, if you know that the developers have a tendency to use nice numbers in their drops (1/32, 1/64, 1/2048, etc), you could use that to have more confidence in your answer.

This is assuming a few things, however:

1. All drops are independent of each other. For example, in the game Overwatch, the odds of receiving the rarest tier of item increases every time you do not receive one. This would already be hard to calculate. Now imagine a system where each tier of item may add or remove the cumulative probability of the good drop. Calculating the percentage begins to border on impossible when you know less and less about the system.
2. All drops are simply based on a drop percentage. For example, in the game Runescape, enemies may have a chance of dropping from the Rare Drop Table, as well as their own drop table. Some rare drops are shared by the enemy's drop table, as well as 'no drop' being shared by both, so without knowing the RDT's rate for that enemy it will be incredibly difficult to predict drop rates to a good confidence level. Other factors may include number of players attacking, items equipped (e.g Runescape's Ring of Wealth), or anything else the developer wants.

You should be able to perform a sample proportion confidence interval, that reads something like "I have 95% confidence that the drop rate lies between 0.79 and 0.81", where your sample proportion would have been 0.80 (in the middle). If this is a bit much effort, just use your sample proportion for a good estimate once you're happy that it's stable. At the end of the day, even if you eventually calculate that probability to be 0.5%, the drop rate doesn't guarantee a drop every 200 kills. Even with 2000 kills, the probability of receiving exactly 10 drops is 12.5%.

Let me try to go through all your questions in turn.

1. Checking whether or not a drop occurs given a certain drop chance

Just use a pseudo-random number generator and compare the result with the drop chance, just as you suggested in your question. For example, if you specify the drop chance in percentage points with up to two decimal places, you could ask

rand = generatePseudoRandomNumber(0, 10000) // between 0 and 10,000
if(rand <= dropChance * 100)

2. Checking which events happens given a set of possible events

Try to use a data structure that is more flexible than hard-coded switch expressions. I coded a basic version of a Lottery class here: https://gist.github.com/3368046. Lottery allows you to do stuff like this:

$Lottery = new Lottery();
$Lottery->addEntry($item1, $dropChanceOfItem1);
$Lottery->addEntry($item2, $dropChanceOfItem2);
$Lottery->addEntry($item3, $dropChanceOfItem3);
$drop = $Lottery->getWinner();

With my implementation, the drop chances do not even have to add up to 100%; the lottery determines the winner based on the number of lots that each participant has in relation to the other participants of the lottery.

3. When to check for drops and how to communicate the result to the player

I'd decide based on what the most critical bottleneck is: If bandwidth is expensive and if your service has a good latency (Amazon?), go for variant 1. If your service has latency issues, go for 2.

Try randomly checking "will this item drop?" for every item. You could store the information for item and drop chance as pairs in a list:

drops = [["sword",5],["hp_potion",35],["mp_potion",35]]
loot = []
for i in drops:
    if random_chance(i[1]):
        loot.append(i[2])
give_player_loot()

...but this does not account for quantity, and maybe you'd like to prevent one item from dropping if another does. My solution for this would be nesting lists even further, with an item name, chance, and min/max quantity going in list A. List A is put within pool B, within the drops variable, C.

So, this would look something like:

# SLOW BUT EASY WAY TO BUILD LOOT TABLE
# You do this when the monster is created, if you are keeping a list of monsters
# that is made at the start of the game, this is also where you'd put this.
# this example loot pool has a 35% chance to drop 1-3 HP potions, OR, failing
# that, a 35% chance to drop 1-2 MP potions.
A = ["hp_potion",35,1,3] # individual item drop chance
A2 = ["mp_potion",35,1,2] # another one, to help describe the concept.
B = [A,A2] # the loot, A and A2 is going into pool B.
C = [B] # B goes into the final drop pool
monster_drops = C
# COMPACT WAY:
monster_drops = [[["hp_potion",35,1,3],["mp_potion",35,1,2]]]

Next, we have to iterate between all drops like so:

loot = []
for i in monster_drops: # go through all pools
    for j in i: # go through all items
        if random_chance(j[1]):
            loot += [j[0]]*random_range(j[2],j[3]) # add this item to the loot,
                                                   # multiplied by the number of
                                                   # items it should drop.
            break # exit the loop
give_player_loot(loot)

Sorry for the poor formatting, it's based on python and I don't really know how to use StackExchange, but I hope this was helpful! (Also, I noticed you are using JS, but hopefully the same concepts apply there!)