Primers are usually brass or nickle plated brass. I believe the mild steel ones are more commonly found in surplus Soviet ammo.
I just checked with a magnet and none of my pistol or rifle primers (CCI and Win) will stick but my CCI 209 shotshell primers definitely have steel sleeves around the cups.
There is lead in primers in the form of lead styphnate which is the priming compound in the vast majority of primers. It isn't metallic lead but lead locked in a compound. Kind of like how blood contains iron but you can't exactly form it into ingots or anything. We're also talking about absolutely tiny amounts of lead; even if you extracted all of it (much of it leaves your barrel when the firearm is discharged) it would form a near microscopic speck of metallic lead. Most of the priming compound will be the 2,4,6-trinitrobenzene-1,3-diolate which is the organic compound attached to the lead atom.
I personally dump my used primers in the garbage but I suppose I could put them in with my scrap brass to try to get some money for them from the recycler.