Obviously, anything dirty or moldy needs to be thrown away. I was helping my dad clean his house, and I offered to take some things to goodwill. He insisted on throwing away things like puzzles, and air hockey pucks, because he didn’t think they had much value. What items shouldn’t be given to goodwill because of their lack of value?
All of the “thrift stores”, of which there are many, always seem to be full of near useless kitsch. Things like jigsaw puzzles, costume jewelry, little porcelain figurines, stuff like that.
They definitely want things in good repair. If you have anything that requires repair or mending, my experience is that they don’t want to take it.
You can never go wrong with letting the folks at Goodwill make that decision.
Yes. When donating clothes, etc. they don’t sort until it gets to the facility. It is on you to not be a dick and donate gross clothes, but they have no way of checking. If you’re unsure, and it’s not broken or disgusting, donate it. Some exceptions should be made for bulky things - they won’t take CRT monitors and TVs, for example, even if you paid them.
Anything still useful and small enough to be gift wrapped and passed around a table can be used to play white elephant.
On the clothes thing - as long as they’re clean, many charities will accept them in any condition of wear; anything not fit for resale as-is may be sold in bulk for rag recycling - it goes off to be further sorted into streams that go into paper manufacture, or get shipped to developing countries to be remade into rugs, bags and other craft items for export.
Ask 'em. Probably varies a lot.
We have a local charity shop that specialises in broken stuff; they train (usually formerly homeless) people to fix things, which they then resell.
The goodwill stores around here say “You can give it to Goodwill if you would give it to a friend.”
I’ve read articles about how thrift stores frequently get inundated with junk that they then need to spend a lot of time sifting through. They tend not to turn away donations because they don’t want to discourage them, but some people use them as a junk disposal service. If you could see yourself successfully selling it at a yard sale (even for super-cheap), it’s probably safe.
If I could use it but don’t want to, it goes to Goodwill or a like entity.
If I wouldn’t/couldn’t use it, it gets trashed.
We have a small charity that aids epileptics that will send a volunteer over to your home and go through your garage or basement or whatever with you and let you know what they’ll take and what should be pitched. They were very helpful to my next door neighbors when they were cleaning out their deceased parents’ house. I’m not sure how many charities would do that, but you could always ask around.
My personal rule is based around whether or not an item is still functional for its intended purpose. So clothing that I wouldn’t wear because it was too worn/damaged gets tossed. The T-shirt with a giant rip where the pocket snagged? Trash. Shirt my mother bought me for Christmas that’s weirdly patterned in green and purple but otherwise new? Donate. If I just don’t like it or it doesn’t fit or I’m thinning the herd and it doesn’t make the cut, then I’ll donate it. Same with small household items. If it’s still functional, it can go to Goodwill. If not, trash.
Since my husband and I are staring into the face of moving to a new house (same city, more convenient location), I’ve been making this decision quite often lately. Although, I have also found (to my delight) that damn near anything I leave on my front porch with a note on it saying “FREE!” will magically disappear within 72 hours. We’ve gotten rid of all sorts of stuff that way. Including many things that would have been seriously awkward to haul to the Goodwill.
True. Although many of these organisations probably get a special deal on their waste disposal facilities, it still won’t be free for them. Donating garbage just cuts into the charity revenue.
Have you been in a thrift store recently and seen the quality of wares they have available? That would be an easy check. If you think your thing would fit in on a thrift store shelf, then it’s reasonable to donate.
Anything good I set out by the trash cans and it just magically disappears.
One of my relatives buys jigsaw puzzles. She visits the local thrift shops every time she passes them, and any puzzle with cats, birds, flowers, pumpkins, cottages, barns, or any of the other myriad subjects she likes goes in her cart. She’s elderly, doesn’t have much to occupy her time, but sits in front of the TV with her latest puzzle. She’s happy, I’m happy she’s happy, keep the puzzles coming.
You don’t have another relative who buys air hockey pucks, though, do you? 
Man, that sucks. Like epilespy isn’t hard enough, and then you get AIDS. 