I have in my inventory a broken grenade. I can't equip it or repair it (really, would any be crazy enough to try that?) and I have know idea how it came to be in my inventory. Best I can think of is that I recently shot the grenade in a SM's hand, ending his torment,and scavanged a 'nade from his corpse. Anyone else seen this?

You are correct

You can repair said grenade, if you ever find another broken one. I have a batch of broken and not broken nades in my locker.

I always find them broken grenades on raiders near some baseball field.

Vendors can repair broken grenades too.

Interesting. I've shot plenty of grenades in enemies' hands before and looted the corpse after, but this is the first time I've seen a broken grenade. I guess it's more common than I thought. BTW, would anyone know this: when you do shoot the grenade, does it do damage according to the thrower's or my explosives skill level? I'm wondering because I've got demo expert level 3, and high explosive skill (75 or more) so if it's based on my damage, shot grenades should be very effective.

I don't think it's based on that. I have all 3 Demo perks and an Explosives skill of 96.... that dude should be particulate matter. I HAVE taken out more than three raiders by shooting one nade in someone's hand, but that was lucky I think.