When you pay from a P2SH address, does its change go into a P2SH address too? Because that means you must always keep the redeem script around to make the change addresses, and if it's a multisignature script, then you'll need to have the script with those public keys around, ready to go too.
When you pay from a P2SH address, does its change go into a P2SH address too?


No. Not necessarily. You can include p2pkh/p2sh/p2wpkh/p2wsh outputs.


  Because that means you must always keep the redeem script around to make the change addresses



If you consume P2(W?)SH UTXOs to produce new P2(W?)SH outputs, you generally keep the redeem script or communicate them well in advance with the recipient (if not yourself).
If you consume P2(W?)SH UTXOs to produce new P2(W?)PKH outputs, you do not need to communicate the redeem script as the recipient can work it out themselves.