Your first question has a long and complex answer, but it boils down to this: You're playing a game where 2-4 bullets are lethal, fired from guns that shoot 600-900+rpm (that's 10-15 shots per second!) Some guns will kill in one trigger pull.
Killcams are not, were not, and will not ever be an exact representation of what occured because what is happening for each player is always subject to the lag monsters. You'll note that there are barely any (are there any?) other fps games out there that show killcams. They just say X kiled you with Y and maybe show you the person who shot you.
Treat killcams as tools to see where a person was moving, how they ambushed you, or what they were using, don't use them as tools to determine why you lost a heads up firefight, that's just going to piss you off.
When you're standing stationary around a corner and someone comes blazing around the corner firing at you, on their end, their client says 'this dude is at position X' and they come around the corner, see you, shoot you, and you die. On your end, your client is still waiting for the position + firing update of the guy that came around the corner. He comes around, you fire and kill him! Then you fall flat on your face dead. wtf?! To quote a great man: "You are already dead". You just didn't know it yet.
While the specifics of how the net code has been handled have varied from cod to cod over the years (getting progressively shittier in every one after mw2 imo), the basics are the same: If you're in a situation where you're going into a coin flip face to face firefight, it's a coin flip firefight. The trick has always been to avoid getting into those situations as much as possible.
That's part one.
To answer the second part.
If you are going to get into face to face firefights frequently, you need perk muscle to survive them, and a weapon with a low enough ttk to beat whatever your opponent is using. Milliseconds matter in close range cod fights.
Relevant gunplay perks: Ready Up, Quickdraw, Focus. Faster gun up after sprint, faster ads, better stability from flinching when hit with enemy fire.
Stalker is also on that list, though it doesn't directly affect speed, it lets you stay fully mobile while ads, which can be stronger than Ready Up and Quickdraw together if used well (if you're already ads and moving around a corner, you are all but guaranteed to be faster than someone who is mid sprint, even if they have ready up and quickdraw both on an smg).
The gun you are using determines mobility and ads times. SMGs and Shotguns are fastest, ARs are in the middle, MRs and LMGs are next, Snipers are the slowest (ads wise, though there's a bunch of variance for both ads time and mobility in that class).
Ultra bad habits that will get you killed 24/7:
Sprinting constantly (Marathon is dangerous)
Reloading after a single kill (push X to spawn enemy). Sleight of Hand and On the Go both can reinforce this bad habit.
Not prepping for the second (or third) enemy as soon as you down the first. There is always another enemy around the corner. They saw you, heard you, saw their teammate die, heard your gunfire, saw the tracers, saw the skull marker on the ui, were in party chat, saw you on the minimap, saw you with an oracle, tagged you with recon, heard you with amplify, whatever. They know you're there when you kill someone nearby. Be ready for it.
And then there's stealth. The combat perks are critical in general, and if you are using a build without some combination of quickdraw/stalker/ready up/focus, you need to know what you're doing, but stealth is equally important.
A silencer on your gun, Off the Grid, Dead Silence, and to a lesser extent, Incog and Takedown are all important for protecting you from the minimap, SATCOMs, and headphones/amplify. Incog and Takedown have more subtle effects, but both are useful. Incog is notably effective at longer ranges by preventing your name from popping up from a casual ads sweep, but it can even save you in cqb if someone hesitates for even a second because they didn't see a red name. I've died a few times myself because I mistook a friendly name behind an enemy as their own!
In terms of enemy positional knowledge, Amplify is the single most important perk to take, because it alerts you instantly to nearby enemies without Dead Silence, and gives you a chance to hear enemies with Dead Silence, something you have almost no prayer of without.
Sitrep is quite useful, particularly in modes like TDM where kids like to camp in buildings with IEDs guarding them. Sitrep protects you from all types of equipment and gives away enemy positions.
Wiretap is potentially strong, but as a 3 point perk and the number of people running Off the Grid, I'm not super fond of it. That said, a build using Wiretap and Hardline with Assault SATCOM (and nothing else!) is kind of hilarious. You'll have near perfect locational info on anyone not using Off the Grid.
Also fwiw, every single one of my srs bzness builds (or even semi serious) are 1 gun, 11 perk points. Equipment is pretty crap in this game with no way to resupply it short of completing a field order, I'd rather have the 2 extra perk points. A secondary weapon does have some value on certain setups (e.g., I'll use a pistol on an lmg build for mobility), but you can often get away without one.
You'll note that in all that, I didn't make any mention at all about your personal aim or the sweet cod skillz needed to survive a close range firefight. That's because they're secondary to everything else. My k/d has been going up over the years I've played CoD, and I'm getting older, slower, and blinder every year. I'm not doing well because I can out 360 noscope the 14 year old kids that make up the cod audience, I'm (usually) doing well because I'm flanking them, shooting them in the ass, killing them with a faster weapon, or killing them at close or long range with a close or long range specialist weapon.
A big chunk of that comes down to map knowledge, mode knowledge, and whether or not you're playing with a party.
Playing cod solo is way, way harder than playing in even a small group, and similarly, certain modes are much more difficult when solo.
Domination is a really good example - in a fully organized team, you cap two points, and have players covering all possible approaches to your two points. You never push into their spawn. As a result, the spawns are stable, you don't get shot in the ass, and it's fairly easy to lock down an area. Worst case they break out and back cap you, you rotate and resume facing the other direction.
Contrast that with public game Domination where players run all over the fucking place, and move across the map in a straight line from their spawn point until they hit the far enemy flag and then do a 180 like some sort of mindless rifle armed pong ball.
In that case, enemies spawn all over the goddamn place, you get shot in the ass constantly, control points flip constantly, and the game is generally a mess. The worst of these matches on some maps just make you want to hide in a corner and not move at all.
Unfortunately I don't have any great advice for that last part. The only way to improve in that respect is to play more, learn the maps cold, and figure out where people move in each game mode, and where people like to camp. This is how you develop the sixth sense that protects you even in the absence of a uav sweep/amplify/etc, and lets you hunt enemies as they spawn instead of being hunted or spawn killed.
And on your last query, the damage is the same as its ever been. ARs kill in 3 bullets, 4 or more at long range. SMGs kill in 3 bullets up close, 4-6 very quickly at longer ranges. Same with LMGs, just a longer distances. MRs can kill in 2 bullets or 1 headshot at shorter ranges. Snipers and Shotguns can kill in one shot. Pistols can kill in two up close, one with an accurate upper chest/head magnum shot.
It feels super fast partly because the netcode isn't garbage in this game, and partly because the way that damage is 'delivered' can make it feel instant - ie in your screen you fire a bunch of rounds into a dude and you know you missed some and hit him 3-4 times, then he died. On his screen he saw you and then exploded instantly. It's just the way lag works.
If you're trying to analyze how they're killing you so fast but you're not, or how you're losing those face to face firefights, you're sort of looking in the wrong direction - you need to back up and look at your perk setup, your weapon, your streaks, and most importantly, your movement and your map knowledge.
Unless your build is intentionally designed to get up into peoples faces and win the fight (shotguns, some smg setups), in most cases you don't want to get into those fights in the first place. This isn't Halo or Quake. You don't get into 10-15 second duels with people. You spot someone, you open fire, they die nearly instantly. It's hide and go seek writ large.
Anyway, that's a lot of words. Cod's a tough game to play well, and every year we get a new cod with new maps and a new of gun/perk/streak balance and people suffer as they have to relearn the map flow and what sorts of builds are effective.
Best advice, play with friends and relax doing stupid shit. If you want to improve your game otherwise, look for some quality streams on twitch and see how the solid players play. Don't look up cherry picked youtube videos, a handful of amazing games can come from hours of mediocre footage (e.g., I have a back to back double loki video clipped that came from a night of play where I'm sure I had plenty of 20-20 rounds and lots of dying around it). See how they play moment to moment - what builds they use, where they move on any given map in any given game mode, etcetc.
edit: And turn your sens down. Only time high sens is a good idea is if you're using kontrol freex or your aim is stellar. High sens lets you spin fast up close, but it harms your mid to long range accuracy, which you need to do well. Aiming up close is easy either way, and spinning 180 will save you/get you some kills occasionally, but not as many as the kills you'll lose from poor aim at a distance, imo anyway!
