Alright, this might devolve into a rant, but I'll try to keep it level.

I had a game today with an Akali on the enemy team doing what an Akali usually does: build AP, sneak about and snipe loners or kill our carries and vanish.

Now, during the course of the game the enemy team continually complained about Akali never being around for team fights, and it hurt them in the long run. The Akali player would use the team fight time to go farm neutrals or advancing minion waves, but never help hit my team or even a tower. We plowed through their team easily without Akali's damage presence, and eventually our Malzahaar killed him/her on his own through the benefit of being in a team fight: lots of assist gold.

At the end of the game, the Akali used the excuse, "Assassin characters never (NEVER) join or go near team fights, because that is not their role." Their role is to (1) farm, and (2) roam and kill.

Well golly gee, that's fine and swell until the end of the game where both teams are in a clump and roaming is dangerous/suicidal. During the team fight phase, an assassin should be nearby (not INSIDE the group, or at least not in front) waiting for the best opportunity to strike the weakest or most powerful target.

I have never played assassin characters like this Akali did. He/she was so adamant about Akali/Eve/Kat doing stuff alone and not being team players that I swear they actually believed it.

Does anyone else believe this to such an extreme? Why/why not?

She should be around for team fights but its fine if she goes off its just a skirmish unless she can really help you. She should just wait around behind the enemy and when they get split or someone gets low just r in and burst them down then get rid of focus then try to kill the next squishy.

When I play Akali, I wait a few secs and then I find their squishiest champion, burst him down, flash out, and run away or pick off people who run away from their team.

Squishy characters should never go in first. Assassin champs are meant to jump in after all the initial nukes are used and kill the squishies.

Akali should always be looking to jump on the enemy carries, but never initiate.

Katarina lives for teamfights. The Katarina ideal is to get a full ultimate off while standing in the middle of the entire enemy team. Get that and you win the game, but it's amazingly hard.

Evelynn functions like an AD melee carry lategame such as Master Yi or Xin Zhao. Run in after all the scary stuff is gone and murder people.

The hardest part about these champs is knowing when to go in. Go in too early and you die instantly. Go in too late and the fight is over so you also die instantly.

That's one massive "..." there. For Akali case, she should be on standby ready to team fight if people start gathering. Not making excuses. If she's not on standby at the team fight, she should at least take a tower while you guys hold them in place. Other than that, an Akali should wait till the fight starts and the initial burst of damage and cc goes down. That way, they should be able to go in and target the threat and neutralize at as soon as possible. Depending on the outcome of the fight, she can continue to the next target or back out if it's truly unfavorable.

An assassin can be essential to winning a teamfight. You don't start it. You may even wait a couple seconds before going in. That's fine. But then you circumvent the enemy team with whatever kit you have and burst their carry down, which can very easily win a fight for your team.

Granted, if the OP's characterization of the game is accurate, Akali was definitely wrong; however, as often as I see people avoiding teamfights and costing their team a valuable player, I more often see people initiating at extremely inopportune times...

When you do have 4 people and the other team has 5, initiating at a disadvantage is stupid...If Akali is across the map, pull back to a tower (gives you a virtual 5th champion). Don't go rushing in to battle. If you manage to stun a character away from their team, don't rush in. Burst from range...if you get the kill, great, if not, LET THEM RUN - even though you didn't kill them, you most likely nuked them enough to effectively remove them from the coming fight. Give it a few seconds before you consider engaging so that your cooldowns come back up. If you fail to kill that champ that was out of line and give chase, you will run into the other 4 people on the other team while your abilities are on CD. You will die. This is the most common mistake I see in teamfights.

Ranged AD carries and AP carries are great for nuking down and punishing someone for being out of line. Tanks, Let them. Don't you dare initiate on this. It's not your job to guarantee kills. That's the carries' jobs. It's your job to keep the carries alive after a fight so that they can knock down a tower or push an inhibitor.

A similar situation to the above happens when tanks get greedy for kills because someone was caught out of place....as a TANK, don't try to nuke that player yet. Wait til your team nukes them, has their cooldowns back up and you can initiate on the whole team at an advantage now...you'll get more gold from getting 4 assists on the main team than getting the 1 kill and 0 assists because you initiated, got the kill and then subsequently your team was aced because everything was on CD.

TL;DR;
As often as someone not being part of a teamfight is a problem, so also is the team initiating before that person can get there.

Don't try to be pro. Just play smart.

As Akali you have to wait until most of their cc and heavy AoE is used or you will be annihilated. Choose a right moment and jump on their carry and rape him/her, after that you just finish the survivors as it is very difficult to run from Akali.
Poppy is different - she is more team independent, can just walk in the middle of the enemy team, kill the carry and run away lol.

no, that is so wrong. EVEN IF akali dies during the teamfight, if she just joined, she'll have accompalished 2 things:

Unload her burst on some unfortunate ashe (dead or not doesnt matter, someone else can pick off that later, or she'd have to retreat too considering her hp is now low), causing the enemy team to lose a damage dealer even if your team loses 1 too

Tanked a tiny bit (even that 1 spell needed to kill her considering she's squishy means 1 less spell on another carry).

i play yi (darned squishy), even after going in the right moment, i eventually get focused dead. but, i bring down their veigar before i do, so essentially i did my job. not being present in a teamfight is definitely wrong, death isnt something to be feared here, its just a game you can revive. Always stay for a teamfight unless it is really really obvious that with/out your help they lose anyway