So, I'm going to write these in order of Metal Gear games (which is not chronological order) because I think that's the way he wanted the story told. I'm also going to start at Metal Gear, not Metal Gear Solid, because if you didn't play Metal Gear there is something wrong with you...
I'll also try to keep this brief
The Protagonist (largely across the series) is Solid Snake. In the beginning you're the Rookie on a team called FOXHOUND which is headed up by a man known as BIG BOSS (these are all call signs). You're goal is to infiltrate a compound known as "Outer Heaven" which is run by a bunch of futuristic mercenaries lead up by the "Legendary Mercenary" (I assume that sounded cooler in Japanese). Your buddy Gray Fox went in before you but they lost contact with him so you have to get in, save him, and figure out his cryptic last message "METAL GEAR."
The game is actually a lot of fun but comes from the school of HARD where Mega Man 1-3 and Dwarf Fortress spent their formative years. Eventually you rescue Fox only to learn that Metal Gear is a giant Mech that can launch Nuclear Weapons. Everyone seems kind of hung up on the Nuclear part... You neutralize Metal Gear only to find out...
...that the "Legendary Mercenary" is your boss: BIG BOSS. You fight BIG BOSS and escape before the place blows up.
A group called Zanzibarland captures a scientist Dr. Kio Marv because he's solved the worlds Oil Problems (why is kind of silly). Solid Snake comes out of retirement (this happens a lot) and joins FOXHOUND again under Roy Campbell. You're to deal with Zanzibarland who you quickly find out is headed up by BIG BOSS; also he has another METAL GEAR model D.
You find out that Gray Fox has betrayed you (betrayal in a Metal Gear game? shocking!) and you manage to arrive too late as the good doctor is dead. You do get the world changing formula and then fight Gray Fox and BIG BOSS. Not nearly as cool as the first game, but the gameplay is smoother.
FOXHOUND rebels and takes over a compound in the Alaska known as Shadow Moses. Once again Solid Snake has to come out of retirement to deal with the bad guys. Luckily this time BIG BOSS is actually dead, but unluckily they have a new METAL GEAR: REX. They also have a couple hostages including Roy Campbell's niece (I never was 100% why all of FOXHOUND except Campbell revolted). You sneak in and save the hostages, fight Revolver Ocelot, who seems to think this is the 1800s, get saved by a creepy Cyborg Ninja, who no one knows and may or may not be a friend and have a short love story with Meryl Silverberg (Campbell's niece).
You fight your way through former FOXHOUND members, including some really cool guys like Psycho Mantis who literally reads your save files; find out Cyborg Ninja is Grey Fox with a Lobotomy; you get caught by the bad guys, tortured by Ocelot, figure out people have been lying to you, figure out you're a clone, meet your clone brother: Liquid Snake, destroy METAL GEAR REX, kick Liquid's butt, then watch him get killed by the Fox Die Virus; find out Ocelot is a spy for the President and that one day the Fox Die Virus will kill you too (but not today).
All in all this game starts off sane and gets progressively less so.
If you thought the plot to MGS was convoluted just wait... There are two stories in this game.
In the first story you're once again Solid Snake, only this time you're working for a renegade group going round and destroying METAL GEAR in third world countries with your buddy Otacon (one of the designers of REX). You find out the U.S. Navy is building anti-METAL GEAR METAL GEAR, called RAY, and instead of giving them a slap on the back you decide to sabotage the entire thing... You sneak on to an Oil Tanker, figure out how to fight (something you could barely do in the last 3 games), only to see Revolver Ocelot take over the ship with a bunch of Russians. Apparently, Ocelot thought it would be a good idea to replace his hand with your brother Liquid's and thinks he's Liquid now... well sometimes he does, it's weird. Liquid/Ocelot sinks the tanker and escapes with RAY.
The second story happens two years after the first. Apparently after the tanker went down they build a giant Oil Platform to cover up the whole METAL GEAR thing; and sure enough it gets taken over by terrorists known as The Sons of Liberty. The newest member of FOXHOUND Snake... I mean Raiden gets sent in to take then out and rescue the President. This is the first game where you don't play as Solid Snake. Now you might think: why not just send in Navy SEALs and let them take care of this? I guess they did that and it failed miserably. So bad there is only one left: Lieutenant Junior Grade Iroquois Pliskin who, for some reason, looks a whole lot like Solid Snake; go figure.
TRIVIA: Snake Plissken, played by Kurt Russel was the main character in the post apocalyptic movies: Escape from New York and Escape from LA.
Cyborg Ninja shows up again (despite dying in MGS) calling himself Mr. X and Revolver Ocelot/Liquid Snake has showed up for shits and giggles. You fight someone claiming to be Solid Snake (who Pliskin assures you isn't), who later turns out to be Solidus Snake (its like they're not even trying...), rescue the President, find out the whole oil platform thing is to hide ARSENAL GEAR which protects a super computer known as GW.
In a plot line right out of Live Free or Die Hard, GW can control all of the internets... I guess. You get captured, tortured, find out Raiden used to be a child soldier and Solidus your adopted father, find out your Girl Friend and Commander Colonel Roy Campbell are both just GW. Apparently, the entire world is a sham and its run by these guys called the La-Li-Lu-Le-Lo (or the Patriots for short). The Patriots want to use GW to know everything about everyone always (if that makes sense) but they don't want people knowing stuff about them, so they built GW to collect all information about everyone but them, so Liquid figures he can use that to track down the Patriots and Solidus just wants to blow it up, and apparently the Patriots set Solidus up so this entire thing would be a simulator for Raiden to become the next Solid Snake. So you beat 25 METAL GEAR RAY, get your Fission Mailed, crash into New York City, defeat Solidus, find out your GF isn't a computer program and generally save the day.
No part of the Patriots story line is concluded, and everyone seems to ignore this for no good reason.
After the cliff hanger at the end of MGS2, in true Metal Gear Solid fashion, MGS3 has nothing to do with it. Instead, MGS3 is a prequel. You're Naked Snake (see how hard was that) who was trained under The BOSS a hero of WW2 who also defects to the Russians. You infiltrate their base, learn about an Illuminati-esque organization The Philosophers who may or may not be dead, meet a young Revolver Ocelot, find out The BOSS didn't defect and is only on a mission, beat up everybody, save the world (but in secret), get dubbed the boss of all bosses: BIG BOSS, and finally get to have sex with a hot chick. Oh and Ocelot works for the CIA under the call sign ADAM... no clue why.
Honestly, it has almost nothing to do with the Plot, but it is one of the best games in the series (on par with MGS or the first story of MGS2) definitely worth playing, but completely irrelevant to the plot. I realize the "Philosopher's Legacy" (a giant pile of money) is supposed to bank roll the La-Li-Lu-Le-Lo, but I could have done without that bit of fiction.
I have a confession, I never played MGS4 (or any other Sony Exclusive game after 2005, I might be vindictive), but instead watched the entire thing on Youtube. If you have 10 hours to spare its not a bad watch. As such, I can't really speak to the game play, but if MGS2 and 3 were any indication it was probably decent.
I've put the entire explanation in spoiler because the original asker wanted to know everything up to 4.
So apparently after MGS2, when no one decided to do anything about the Patriots they took over the world and made it all about war, either your economy was based on war or your country was in a war. But since it's a modern war its all about the genetic engineered soldiers and nano machines. Controlled, of course, by a system called Sons of the Patriots or SOP (weren't these the guys trying to hide their existance?); and Liquid/Ocelot is out hijack it. So sure enough, Roy Campbell asks Snake to come out of retirement, one more time, and deal with Liquid/Ocelot. Hey, it could be worse. You could be Raiden again...
Snake decides to start looking in the Middle East (probably because this game was made 2006-2008) and it looks like everybody and their mother has METAL GEAR. Sure enough he runs into Liquid's forces (read: Cyborgs) and catches a glimpse of Liquid along with Naomi (person responsible for FOXDIE, which seems to be killing Snake these days). Liquid has been testing his SOP destabilizing machine which seems to work on everybody (nanomachines) except Snake's buddy Johnny Sasaki.
After escaping Naomi tells Snake she's playing along with Liquid and to come to South America. Apparently the Patriots are controlling SOP (shocker) and Liquid still hates them. Snake chases Naomi down in South America and she tells him Liquid wants to get BIG BOSS' biometric data so he can control SOP. She says that BIG BOSS was one of the Patriots, but that's hard to believe based on the first 2 METAL GEAR games. She also tells Snake that he's dying from FOXDIE, and he's infectious to other people. Naomi get's captured again, you track her down again and you meet up with Raiden, he's a cyborg now...
After getting out of South America you head to Europe to try and save Raiden's life (he's not doing to well), instead of trying to stop liquid... or the Patriots... or anything else remotely useful. On the plus side apparently BIG BOSS' remains are in Europe, so I guess you can keep playing the game. Probably would have been better if she mentioned that before hand... While in Europe you meet up with BIG MOMA (I'm not making this up) who was EVA back in MGS3 where she had a bit part not worth mentioning other than she's Snake's mom. She's got BIG BOSS' remains so Snake teams up with her, only to get attacked by SOP. After a pointless battle Liquid shows up and destabilizes SOP again for an easy win. He then destroys BIG BOSS' remains... Wait, what? Didn't he want those for some reason?
Turns out Liquid is planing on destroying the SOP core with METAL GEAR REX and replace it with his own. I guess BIG BOSS was a Red Herring. Just for old time's sake, Liquid is planning on launching the Nuclear Missile from METAL GEAR REX off of Shadow Moses in Alaska (MGS1). This turns into a nice trip down memory lane with Cyborgs replacing the original FOXHOUND members, only this time you get to pilot METAL GEAR REX and fight METAL GEAR RAY in GEAR ON GEAR BATTLE ROYAL!!!. Both METAL GEAR are destroyed and Liquid reveals he has a Submarine named Outer Haven (after MG) which he's actually going to Launch the Nuke from.
Apparently, Liquid has already inserted his SOP Core into the main SOP Core (which is how he destabilizes it all the time) and he just needs to destroy all the other ones to control SOP. Snake decides to sneak into Outer Heaven to inject a computer virus. You fight Screaming Mantis (who is actually a product placement for the PS3), Johnny gets your love interest Meryl, Raiden loses his other arm (he lost his first during GEAR on GEAR), you upload the virus called FOXLIVE (get it guys? its a play on FOXDIE...), destroy the SOP, save the world from the Patriots, and have a final fight with Liquid/Ocelot where you replay the final fights from MGS1-3.
The Epilogue is an hour long... Meryl and Johnny get married, Raiden meets his son (with Rose), and Solid Snake shoots himself in the head so FOXDIE doesn't consume the world... No wait, he doesn't because he's stopped by BIG BOSS who never died. I guess the body you burned in Europe was Solidus. Apparently, the Patriots were BIG BOSS, Ocelot, EVA and a couple of their buddies from MGS3. Everything was going great until BIG BOSS decided War was awesome and the Patriots splintered. The other half of the Patriots wanted to control the world through machines but BIG BOSS was all about the Soldier. BIG BOSS wanted to shut down SOP because it also controlled the life support to the other faction of the Patriots. FOXDIE was only designed to kill BIG BOSS, and Liquid and Solid suffered from it because they were clones of BIG BOSS, but no one else will ever get hurt. Then BIG BOSS dies, and Snake gets to live happily ever after.
Also Ocelot was never possessed by Liquid Snake, BIG BOSS just hypnotized him into thinking he was...for no good reason.
Ok, now reread the whole thing in Yatzee's voice (from Zero Punctuation).
Best Answer
I don't believe there are any overt game bonuses for playing lethally vs. non-lethally, but there are many intangibles based on playthrough. There are badges you get at the end of the game based on how you played, for instance, whether you destroyed everything in your path or were silent and stealthy. There are also specific things you can obtain if you defeat certain bosses non-lethally. (Again, with no actual bonuses.)
It's been a while since I played, but from what I remember, if you put them down non-lethally and got yourself far enough away before they woke up, no help would get called, or at least it would get called to a place I wasn't at.
As to why it's there...it's to let you play in the style you wish.