Time travel and parally universe logic in this show

1. they did not control where and when, there's Fate at work here i.e. yoyo kid, he explains that the flute will only get you to the point in time where you want to save yourself. that's why Lee Gon arrives only after his father is dead, but in time to save his younger self, while Lee Rim arrives before the coup, but fails to save himself from that action. 

2. it's simply the way the timetravel works in this show? they update their memories, and no, it didn't already happened in the original timeline? i don't get why people find this so confusing?

3. yes, and it was explained in the show, when lee gon has both halves of the flute after capturing lee rim, the portal is no longer with a single exit but infinite one.  They were confused, the flute choosing where and when to get them, hence Lee Gon has to check different universes to get to the right one. I don't like the infinite universes thing too, tbh. I would have preferred a single parallel one. but the writer had other ideas... 

4. I have no answer to this :D

I love quantum physics and its related fictional uses, so I'll answer your questions based on current knowledge, theories, "common" fiction tropes, and the canon rules of the series.

>>> If time travel was possible, how did LG or LR control when they went back to? How was LR able to go back to well before the coup started (when he met older LR in the other room)  but LG always ended up arriving few mins before coup/attack on younger LG? The show mentions something about desperation and feelings controlling it. Couldn't LG have been desperate enough to arrive about an hour before everything started? Couldn't he have arrived a day before and warned everyone/convinced at least one person like Lady Noh?

To understand this, parallel world and time travel are not actually different from each other. In quantum physics this is called "spacetime" (one word; inseparable) travel. "Parallel world" travel is moving through the spatial dimensions (x,y,z) like when we're walking, running, driving a car, and so on. "Time travel" on the other hand is moving through the time dimension--naturally we are all time travelers, we are moving "forward" in the time dimension.

Another way to explain it is, "parallel world" is moving horizontally while "time travel" is moving vertically.

The canon rule of the series are the following:
a. If the flute is broken, it is only possible to travel horizontally. Meaning, they can only travel spatially or between worlds.
b. If the flute is broken, since quantum physics stated that "spacetime" is inseparable (can never be separated), then it means time is breaking down. In the series this was presented as the exponentially growing "time stop", which will eventually lead to unimaginable length of time stop.
c. If the flute is whole, its ability to travel in "spacetime" is complete. It can allow traveling vertically or through time (past, present, and future). This was depicted in the final episode where they were traveling both in another parallel world and different time.
d. This is the fictional part: time travel is triggered based on the strong desire (or wish) of the person who used the flute.

As for your question on why they did not travel back in time much earlier, two things:
a. They probably did and it was not shown. In creative writing, scenes that does not add to the value of the story must be removed. In shows it's called "dragging".
b. The characters were afraid and they should because a simple change can drastically change everything … which again the series established. So while they were "desperate" to change things, they were still very careful. Both characters understood it, especially the king.

>>> The memories forming later for JTE and the alternate timeline from ep14-15 make no sense. LG remembers the other LG that came and saved him in the coup. It has already happened in that timeline. If LG was saved in that timeline by this LG, then this LG has already met younger JTE too. Why didn't she remember it until it happened. By the same logic, wouldn't LG actually remember the coup only after he did it?

This is a common confusion in most time travel stories because of what we call the "grandfather paradox". There are many theories on solving the "grandfather paradox" (the one you described) and the most logical and easiest to understand is this: time is both linear and non-linear or to put it another way, it is relative.

Yes, they already met the king in the past but from their vantage point, that did not happen yet so their memories of that past did not exist yet. Their "new" memories only catches up to them once the king actually do the things he did in the past.

In this particular series, there is no "alternate timeline". It chose to go for the theory that there is only one timeline ever.

I hope I explained this good enough, it's better to have a drawing or something while explaining this. ^_^

>>> Was getting to many different parallel universes possible only with a whole flute? It was never really explained. Why weren't LG and LR ever confused about which world to go to? Why did LG have so much trouble in the last episode? It made for some cute scenes but the fan service completely ruined any sense of logic the show was using for parallel universe travel. And honestly a stranger standing in my yard and talking nonsense would be kicked out much faster in real life.

The canon rule of this show is, yes, if the flute is broken it can only connect to one parallel universe. You might be asking next "then why 'that' parallel universe and not the other". Most will probably tell you "literary freedom" or "technical difficulty" but I beg to differ. Here's why - the show hinted which world it was last connected to before it was broken. Can you guess who traveled into the "Kingdom of Corea" universe?

If you guessed the Elder Lady of the court, then you are correct. After she was saved from the raging Korean War in 1957 (if I remember the date correctly), the flute was never used again (as far as clues are concerned). So when the flute was broken, it can only connect to the last world it connected to.

When the flute was whole again, the king had trouble in finding the correct world because this time he has full control of the capabilities of the flute -- that is, to travel in spacetime. He never knew how to use it properly because when it was broken he was just going to the other door. Once its whole, he had to learn to control it first. Then second, he needs to find that partiular universe and remember how to open the door there.

As for "a stranger standing in …" cultural differences. In many Asian cultures the scenes where he was standing was perfectly fine, especially someone holding flowers, with an exotic horse which obviously looks expensive, and a very handsome guy from the heavens.

>>> Since time doesn't stop now when they switch worlds, LG and JTE are actually getting older with time. So they are actually going to travel and meet on weekends until one of them dies? I thought that was implied in the last episode but people here seem to assume they will live forever. Why is the title Eternal Monarch if they'll die?

Theories:
1. "Eternal Monarch" because the king's aging process slowed down very much similar to what happened to his uncle.
2. "Eternal Monarch" because he has access to all spacetime. He can simply travel forward in time in any universe and he's still "the king".

But again, in the last episode it was showed that they grew older at the same time and that is canon. Wanting them to live forever is wishful thinking because the show itself established that they grew older and were still traveling. In other words, he is "Eternal Monarch" because he has the ability to travel in any universe and in any time not because he became immortal.

In summary, nothing is confusing. The writer researched quantum physics well and definitely looked into all the other parallel world and time travel stories everywhere. I actually love how they used current knowledge, theories, and fiction tropes in this show, it's not easy but they did it well.