Do you have any other recommendations for ways to approach FPS multiplayer games if you're a (let's say slightly) below-average player?
I would say a good idea would be to choose carefully which games you go for. I'd avoid the Call of Duty series like the plague, simply because those games reward fast, twitchy aiming and your ability to fire rocket launchers. (And/or shout slurs at people who are better at firing rocket launchers than you.)
In contrast, games like Team Fortress 2 don't rely purely on aiming skill; there is a lot more teamwork involved (in theory, at least...) and you can be very useful to your team without firing a shot, as medic or engineer. Also, in general there isn't much of a skill level expected of you in TF2 - it recently went free-to-play so there are a lot of new players anyway, and it's not assumed that everyone will know what they're doing.
I'm older than most of the gamer community, so I don't have a lot of friends who are playing games online at the same time I am. On some games, like the old Burnout Paradise, I made friends online. But when you're not that good or at least your stats are not that good, people aren't always anxious to see you join their team.
In TF2 there is no way to know anything about a player's skill level other than watching them play (or making a guess based on whether they're wearing a hat...).
BF3 is slightly more ambiguous in that you can see what rank people are. That said, in general I've been playing on 64-man servers which are mostly full; if someone gets annoyed that out of 30-odd people, your team has one level 1... Call them out for being the crazy idiot that they are.
That said, if you don't think you're that good, or possibly don't know what you're doing yet, avoid taking up the "important" roles, which in BF3 are usually helicopter pilots and tanks; both are generally quite a scarce resource in-game and can make a large difference to how the game goes. Also, there's nothing quite as depressing as spawning to join your squadmate as he/she nosedives a helicopter into a mountainside.
I've been playing TF2 since it was in beta, and plan to be playing a lot more BF3 (and play quite a few other multiplayer FPSs here and there). Feel free to add me on Steam or on BF3's infernal Battlelog system if you like (my username's Esvandiary on there too). I'm in the UK so depending on where you are the timezones might only match up if I'm still up at about 3am (which, uh, clearly never happens) - but I'm happy to help you get the hang of TF2 by healing you lots and giving hints, if you decide it's something you want to get into.
Anyone care to take a swing at the weight? d:
I'd rather not; I suspect it'd hurt my hand quite a bit upon impact!
Lead me not into temptation... I can find it myself.