I have thought about these things. I guess I might as well try to lay them out.
DO NOT go the "Run & Gun" route. If all I'm going to be doing is wandering around and shooting things in the face... Well, I'll go play something else. My favorite FPSes so far have been 007: NightFire, Halo: Combat Evolved and Half-Life. Why? What do they all have in common? They all let you do things BESIDES shooting guys in the face. You've got missions, you collect stuff, you solve puzzles, you defend other people, you explore... Make it a nice mixed bag. You don't see platformers only have you jumping around from platform to platform, do you? Not successful ones, anyway. Speaking of which...
It's best to try and avoid most platforming elements. That is, unless you allow the option to play the game in 3rd Person View, as well as 1st. See, 1st Person View is designed specifically for aiming at things. Therefore, it works great for shooting, but, you can barely see your character's body. Not even their feet. That is, unless you look down, in which case, you're not looking the way you should to progress. 3rd Person View works great for platformers because you can see your character's whole body. However, since they're blocking the center of the screen from everything behind them, this camera view is not very good for aiming at stuff. So, know what type of camera system you're using and use it the way that feels natural.
Don't try to be too realistic. Just... don't. We've got way too many games that think they're so realistic, when they have nearly everything displayed in brown and grey, there are walls for cover everywhere and, worst of all, cowardice is rewarded by regenerating your health every time you stop to take a breath. And, yet, you can only sprint for about five seconds? What gives? Half-Life holds some degree of realism, without going overboard. There's enough to make sense, but it also tries to make sure you're enjoying yourself. You don't need to go out getting rations or whatnot and it works just fine. Unless you're trying to make the next Far Cry or, even more extreme, Arma, then, I suggest you avoid any attempts at high levels of realism and not try to be the next Call of Duty.
Allow for customization. Not character skins. It's going to be rather pointless. What you can do is go a route like Serious Sam 3 did. There were five blood modes. There was the standard "Red", a "Green" function, one where you could turn it off altogether and two more where you could replace the blood with either flowers or else candy and sparkles. Now, that's fun! There were also color options that let you play around with the depth and saturation. You could even play in black and white, if you wanted. Though, many people agree there should have been a function that allowed you to play in black and white, but with the blood still in full color, kind of like MadWorld. Also, a content filter helps. Like the blood, you should be allowed to block out language. There probably isn't much else that would need a content filter. But I hear people being rather upset about how their experience is disrupted by all the F-bombs going off every few seconds when they just want to hear some real explosions around them.
Know your weapons. Are you only going to allow a limited amount of weapons on your character like modern shooters usually go for? Or are you going to go like Duke Nukem and Serious Sam by allowing the player to hold every single weapon the game has to offer? If you go limited, don't limit to just two. That is one of the stupidest plagues of modern shooters. You can easily hold more than two guns on you in real life, after all. It's not hard. At least let them hold up to four guns at once. Make sure the weapons are different from each other. Don't have a bunch of repeats. It just eats up extra efforts. If you want to have slightly-different guns like this, allow them to be upgraded and/or modified, like in the Ratchet & Clank series (not an FPS, but a shooter/platformer series). However, going this route is complimented by the "I can hold every single gun" route and really doesn't work if you're going to have to drop one of your guns when you find something you like better. Bombs, grenades, melee weapons... These all work well, too. They've certainly made themselves at home in the genre. So, be sure to include these. Even if your only melee attack is smacking people with your elbow or punching them.
Know your enemy. Or enemies, rather. They can certainly kill you. But, stop making them so weak. They can often be taken down in one shot, while the player can take dozens of hits and still go on (even with regenerating health). However, headshots can really take care of this. Especially, when you're lurking in the shadows. Enemies shouldn't always know where you are. But, they should probably be aware you're around. Just completely in the dark about when you're going to show up and where. They really shouldn't come after you unless you're actually within their line of sight. Or if something mysterious tips them off like the hear a gunshot or someone dies and they happen to notice it. (I'm looking at you, 007: NightFire!) Also, if you really want to go anti-realism, you know what's more more satisfying, humiliating and fun than teabagging? Back in my day, we had this little thing called "pumping so many bullets into your corpse that you explode in a flurry of blood, gore and bones!". If it suits the game's style, make enemies explode. Seriously, the older games had this better. Serious Sam 3: BFE and the fan-made Quake III Arena mod, Generations Arena are excellent examples of this.
Also, you need some form of a "BFG". You just need it. It has to be hard to find and become center stage when someone's got it. And everyone's gonna want to kill the one holding it so they can get to use its firepower before it's burnt out. You've gotta have the "BFG".