I don't see anything about this in the FAQs. Can we upload the free content (wallpapers, artwork, etc.) onto other sites? Or is that against the rules?

I look at the free content as perks for buying the game. Just like you'd get a manual for buying a new boxed game, GOG supplies you the manual and whatever relevant 'perks' as a bonus from purchasing the game.
To be fair, a lot of this content can easily be worth the $5.99 itself if you're a buff of the series and wanted a copy of the soundtrack or whatever else.

Except this content isn't free you had to pay for the game. It's not an 'oh since you bought the game here's this..'
The extra wallpapers, soundtracks, and manuals are all a part of your purchase. That is what makes Good Old Games superior, is that you buy packages, not just the game and then they charge you an extra twenty bucks for the sound track, and the wallpapers and etc. They include it all in one nice, neat, purchase.
I say, distributing any of this content, especially to non-beta players, is EXTREMELY questionable, and to be honest, if I were GoG, I would retain the rights to close your account and cut you off from their servers, if you were found guilty of piracy.

I was sort of wondering this too. I liked the Fallout wallpaper I got with my purchase of the game and wanted to show it off. But since there's a GOG copyright marking on them, I didn't.

Valid... I meant 'included' content by my context.

One more question: is it all right if I just give a wallpaper to a friend if he wanted to use it, but already had or didn't want to buy the game (or even if he plans on buying the game in the future)?

I'm going to go out on a limb here, and say "No."
Point is, if you paid for it, it's yours, but if you give out anything that you paid for, duplicate it, and just give it away for free, you're basically competing with GoG.com at a deal they cannot beat.
They can't give away the games for free, I'm sure if they had enough money and didn't have families and lives, and their own goals and aspirations, they'd probably all love to give these things away for free, if they could get the licensing from the developers to do so, of course.
Their customer service, the games intellectual property (wallpapers, soundtracks, games) are all a part of GoG.com's product. When you start giving away their product for free, you're going to make other people less interested in paying them for their service, which in turn will make it so they can no longer sustain this service (games = money = more games = more money, etc), if they can no longer sustain this service, YOU (read end user) no longer have the ability to download, play, and distribute such a fine quality of service that GoG.com provides, and when you are unable to receive it, so will those who you were servicing with their goods will no longer receive their ability to get it.
What I'm saying, is that if you give away this content for free to a friend, you're cheating GoG.com out of much needed business. They aren't some giant gaming company who's never gonna miss out on if they lose a dime here or there, they're people just like you and me who, despite doing something they love, at the end of the day still need to bring home a paycheck.
Giving away content you paid for, is like stealing food off their plates. Certainly, we would hope they would be paid more than a few dimes, but the fact remains piracy could kill this service, and in order to stay DRM free, I think it's best if we all agree to stop piracy before it starts.
If they can't trust us, what makes you think they would?
My line of work, isn't as tangible as an actual physical product, I actually do customer service over the phone (Technical support), so an analogy using my own line of work would be awkward, but lets say you buy a hotdog from a hotdog vendor who buys hotdogs from a slaughter house in the midwest. If you could magically copy hotdogs and start giving them away, not only would the hotdog vendor be put out of a job, but so would the slaughter house.
If your friend wants the wallpaper so bad, tell him to buy the game, it's only six bucks, and he gets more than the game. If your friend doesn't want to buy the game, give him the six bucks to do it yourself. These games are cheap, there is no need to pirate them.
I mean, the bargain here is absolutely huge, this service, on top of the games, is well worth over (in my uneducated opinion) $80 per 'package.'
You wouldn't download a car, so why would you steal a picture?
I mean, if your friend says 'aw man, it's just a picture, why can't I have it?' tell him 'dude, it's only six bucks, if you're really hurting that bad, I'll give you six bucks and you can buy the game yourself.'

That's what I figured. I told him to buy the game himself.