Okay, I'm going to try and explain what's going on and why. I'm going to address more of the major points/issues you're talking about, so this will likely be a long post.
Let's start with why you have the 20 minute LPQ punishment. To begin with, in order to reach the 20 minute LPQ you have to first leave enough games to receive the 5 and 10 minute LPQ punishments, and then leave enough to receive a third punishment, which is the 20 minute one. So, knowing that you have a 20 minute LPQ shows that you have left quite a bit of games in the past. You didn't get it just for this one particular leave, but this one plus all of the others you have had in the past. That's how the punishments work. Also, the reason why remakes count as left games towards Leaverbuster punishments is because you're still wasting the other players time. Granted, it's not a whole lot, but you are still wasting 3 minutes plus however long it took to wait through the queue and get through champion select, and let's be honest, it can take a decent amount of time to get a game started.
As far as lowering the punishment goes, as DTN said, it is entirely possible. It's done by playing through a certain amount of games without leaving, disconnecting, or otherwise being flagged as AFK by Leaverbuster. The exact amount of games needed for this is unknown, but as DTN said, playing through 10-15 games is definitely not enough for this. You will need to play through a lot more games than that, and I mean a lot. When you do lower your leaver level, though, you will receive a notification about it.
Now let's talk about the individual leaves themselves. For a lot of people, there will be unforeseen circumstances that pull them away from the game. It's unfortunate but understandable. However, Riot cannot really verify if the leave was intentional or if it was due to a personal emergency (such as something you have to do right away or the internet/power goes out), so they have to treat each leave equally. Now, Riot does understand that real life does take precedence over a video game, so to help with the whole "I left due to a personal emergency" situation, they designed Leaverbuster with a but of leniency. You won't get a punishment for your first one, two, or maybe even three leaves, it's when you show a habit of leaving that you get punished. Of course, once you have received a punishment, you have already used up the leniency that the system was designed with, so it's entirely possible that one or two more leaves can trigger the next punishment, especially if they are within a close number of games since your last punishment (again, 10-15 games is rather small).
As for the reasons behind the leaves, well, as I said before, you won't get a punishment for your first one or two leaves. So, if you have the occasional personal emergency, whatever it may be, that forces you to leave, you won't be punished. However, if these personal emergencies cause you to leave enough to trigger a Leaverbuster punishment, there's pretty much always something you could have done to prevent the punishment. For you, you say accumulated so many left games to get you to a 20 minute LPQ (remember, it does take a lot of left games to get here) because of unforeseen circumstances, such as power outages or problems with your ISP. Well, if that's true, then I really hate to say it (trust me, I do), but perhaps it might be best if you took a break from League of Legends. I know it sucks, it really does, but if power outages and such are happening frequently enough to get you to a 20 minute LPQ, then it may not be the best time to be playing League. Instead you could play other games that don't require the same commitment as League of Legends until those things seem to be more stabilized.
As a side note, for this particular instance, you mention that your teammates were in a party that you made, and that you notified them of your absence and they were okay with it. Well, if that's true, then I have a question for you. If your teammates were okay with it, then why did they choose to remake the game?
