If you want to get cotton and no poly, crease the fabric with your fingernail. If it stays creased, it is probably cotton. A few cottons won't stay creased. You can also do a 'burn' test. Burn a scrap of the fabric. If it melts it is synthetic. If it burns it is natural. Wool leaves an ash you can crush - synthetic leaves a solid bead. be careful. If you read the tags enough you will learn to tell the differences in the fabrics by feel most of the time.
I gotta tell you all... I used to do a lot of burn testing. So someone gave my DS (9YsO) a cannon fuse. You guessed it, one day I smelled something so I asked him if he smelled something. No, he didn't. Well, I DID. I started checking around. He had tried to burn test the cannon fuse... He couldn't get it to go out so he finally threw it in the toilet... but not before he left a trail of burn places in the carpet. So even if they aren't small, don't let the kidos see it.
You bet I re-purpose. Been at it for years. I love to re purpose blue jeans into bags, skirts, quilts, what ever. When my daughter was little we didn't have much money so I got a pair of holey pants for a dime and sewed kittens - looked like they were climbing all over the legs and covered the holes. Then I got sweat shirt for a quarter and sewed more kittens on it. She was a hit. I have done the same for the grand daughters with hearts on theirs. Good way to reinforce the knees.
We do historical camping, I used to make a lot of stuff out of old linen table cloths before they got popular and hard to find or too expensive. I used a cotton shower curtain to make a shirt once too. When you can find a pair of linen curtains for a couple bucks - it's a lot of material - why not. Linen is comfy to wear.
Sheets have a LOT of material for the money but some of them have funny texture don't they?