Can I use lumber that has been stored outside?
You may have purchased your lumber at Lowe's "indoor lumber yard" or Home Depot's equivalent, but trust me, they store it outside before they move racks of it inside for sale. Also, the lumber mills where it's cut store it outside. Most lumber yards actually have outdoor storage, too.
To be fair, they will store it on pallets, ensuring the wood on the bottom isn't in direct contact with the wet ground, and usually the storage will have a roof over it, and each individual pallet will often be wrapped with some sort of water resistant wrapper, but they're still stored outdoors. Local lumber mills often won't have any sort of wrapper over their lumber.
Should I put the wood back into my basement?
If you can. Your basement is, most likely, much more stable temperature & humidity wise than the great outdoors. It's also roughly the same temp & humidity as the rest of the house, so after a few weeks of acclimation, you can just use the wood to make a project whenever you're ready to do so, instead of having to plan ahead to let a small bit acclimate.
Has the wood been damaged due to its exposure to humidity and temperature changes?
Only you can tell that because you're the one there. Wood will, inherently, age when exposed to weather - just look at your deck boards (or those of a neighbor if yours are composite wood). Whether the damage is too much for you to use can only be determined by you.
The two most likely to be damaged beyond use are the plywood & particle board, especially the particle board. If that stuff got wet, you'll know it! You might be able to cut the damaged bits off, but you'd probably need to use a circular saw, not a table saw for that because it'll swell and won't be safe to cut on the table saw*. Plywood will also swell if it gets too wet, but it will probably survive more water exposure than the particle board.
The solid lumber should be fine, maybe with some surface weathering on exposed sides/ends. I'd be most concerned about the boards on the bottom. If things were stacked directly on the ground, the bottom layer might be shot, might not, but again, you have to make that determination because you're the one that can see it.
Should I worry about mold?
Do you see mold? If not, no don't worry about it. Man, I think to world is more afraid of mold than the Coldvid. If there's a moldy board, well, there are 1000 questions here about how to clean it up, or just toss it in the burn pile and/or ignore it and use it if it's otherwise solid.
Is it still useable (sic) for my indoor and outdoor woodworking projects?
Only you can determine that because you're there and can see it. You haven't shared any pictures with us, so we can't tell.
Consider this, though - what if you'd made a project with this same wood for outdoor use and it was sitting outside as a completed project as long as this unused wood was sitting outside? Would you consider it junk and throw it away because it's been outside for 12-18 months, or would you just go about merrily using and enjoying your completed project, with the occasional refinishing and/or tightening of bolted/screwed joints?
Yeah, I thought so. Quit worrying and enjoy your woodworking!
*It's not safe to cut swollen particle board or plywood on the table saw because as you cut, the non-swollen part will want to fall down to the table, while the swollen bits will keep the kerf up in the air. As the flat part is falling, it will pinch the blade and a pinched blade will kick back.
A large piece of sheet goods will be pretty heavy and may not kick back much due to its mass, but it can still pull your hands toward the blade (obviously potentially a BadThing™), or shoot the board back at you.
Waaaaay back in the dark ages, when schools still had shop classes, I walked into mine and saw a 2x4 dent in the chalk board at the front of the class. We asked what happened. The teacher said that someone was ripping a 2x4 and it kicked back. The board flew at least 30 feet, over all the student desks and left a large dent in the chalk board. It's a good thing nobody was walking by at the time or he'd have had that dent in his head.
If that's not enough to concern you, go search YouTube for "table saw kickback" to see the joys of what that looks like. I've seen videos where they intentionally generated kickback and filmed it at 25-60,000 FPS just to show how it happens and what happens to the missile piece of wood.
