Short answer: the disc needs to play, the case needs to match it, and box sets need every disc present. Scratched-beyond-play, missing cases, and VHS are the most common reasons shops turn media away.
Music Magpie and CeX both buy CDs, DVDs, games and some vinyl for cash or credit, even in bulk — a good option if you have a large collection to clear in one go rather than trickle into a shop. Broken plastic cases can usually go in household recycling; check your council's guidance, as acceptance varies.
Check the shops currently accepting CDs, DVDs and records near you before you box them up.