[RPG] How to handle character death early on during the session

First and foremost, a GM should always remember that the objective of a game is to have fun. The thing is that "fun" can mean different things to different people, and it sounds like what's fun for you to create isn't as fun for the players when it's executed - and the end result isn't all that fun for you either, since you're sharing this issue here.

Here are my suggestions:

1. Create expectations for the game world before you begin play. By talking to your players and letting them know you will not be pulling punches nor delivering "level-appropriate encounters" exclusively, they are more likely to on their toes and less likely to rush into danger because they (as players) KNOW that there is a possibility that this particular adversary is too strong for them. If this happens, though, expect to see a decreased reliance on combat resolution on behalf of the PCs.
2. Treat the players actions with the same degree of respect you expect from them. A player can only control his or her character in the entire game world. If they do something that should trigger a response, make sure your NPCs respond in a way that is congruent with the setting and with the knowledge that the NPC have of the actions (as opposed to the knowledge that you, as a GM, know). For instance, if your players kill off a guardsman but do it without leaving obvious traces, consider how far the authorities would be willing to go to track down the killer and respond accordingly. Ever played a videogame in which you kill off an NPC in a secluded area of the game world and cops/guards immediately show up guns blazing even when no one SHOULD even know a crime was committed? If your players feel like that, they have no reason to be careful, since they feel the game world will just react according to the GM's whims.
3. Danger is fun; immutable doom is not. As a GM, you always have the power to kill your players, but there are ways that are fun and ways that aren't. Since the job of a GM is partly to provide entertainment (for himself AND the other players), make it a point to find ways to throw threats at your players without instantly killing them. "Lightning strikes you. You die" is a perfectly legitimate thing for a GM to say. After all, he controls the world, which includes weather. Is that fun for a player, though? The example with the assassin that you mentioned, from the player's perspective, amounts to pretty much the same thing. As cool as the concept of the undetectable assassin is, the assassin missing his first shot (or perhaps not scoring a killing blow) makes for a gaming experience as the PCs scramble (they don't know if more shots are coming their way), try to find out who fired the shot (and probably fail since your assassin has escape plans), and are generally spooked because the situation just demonstrated that a) someone is out to kill them, and b) they're good enough to get away with it undetected. This puts the ball in the players court, as they decide what to do about it, and lo and behold, you have a game.

As a final comment, I'd like to recommend Robin's Laws of Good Game Mastering. This book has plenty of tips and hints on how to make your sessions fun for both yourself and your players. It's a resource you might find useful to liven things up around your gaming table.

You can always play in a lower lethality game, but in the question you note that you and your group want this kind of gameplay. So here's some options to keep a player involved assuming character death is present.

1. Have them work on their new character. This keeps them involved with the game and allows you to slot them in potentially later in the same session, depending on time remaining. It has to be done anyway, and they get to still see the gameplay - in a 6 person party you're not doing anything at least 5/6 of the time anyway, so it's not really that big a deal.

2. Give them a NPC in the party, or even a foe, to play. Again, maintains involvement. Depending on how far into the session you are, it might not be for that long. Most of our campaigns have a variety of cohorts, NPC hangers-on, etc. (Sometimes to a fault, our Jade Regent game has a lot by design and our pirate game has two pirate ships full of crew plus more NPCs of PC level than there are PCs...)

Of course if you know you're running a lethal game, have players thinking about a next character in the back of their mind, and you should make sure and have opportunities in the narrative to slot in a new character frequently. I know I don't like to have new PCs just 'teleport in' so having realistic points where a new character might join requires some pre-work to not break immersion.

You could try to "time" major combats towards the end of sessions - but a) people generally time it that way anyway and b) it has little to do as to when someone snuffs it. I remember in our Rise of the Runelords campaign, we assaulted Hook Mountain and the putative Lord of our new Keep got murdered in the first round by a crit from an ogre hook. Unless you're fudging anyway, you can't count on timing.

As a player, I invest strongly in my characters so when they die I definitely want a cool-down period. Playing an NPC or foe or just watching the rest of the group finish up the session at hand while I turn over new character ideas is a good opportunity to do that; maybe I even want to leave early to go have some alone time to get my mind right.