I don't know if you use or have used Bluewallet or Sparrow, but when you scan an xpub (to create a watch only wallet for example), then it shows you 3 attributes: the xpub, the master fingerprint and the derivation path. However, you are right about the derivation path.
I've not used Bluewallet, but with Sparrow it is not necessary to back up the fingerprint. If you are entering a seed phrase for a cosigner then the master fingerprint can be derived, and if you are entering an xpub for a cosigner then you don't need to know the master fingerprint.
In terms of derivation path, Sparrow uses the following for it's multi-sig wallets which conform to both the BIP45 and BIP48 standards, so no need to back them up:
Legacy (P2SH) - m/45'
Nested Segwit (P2SH-P2WSH) - m/48'/0'/0'/1'
Native Segwit (P2WSH) - m/48'/0'/0'/2'
Earlier you suggested 2-3 or 3-5 multisig. Even though I understand that it's the common use, I don't really understand why not using 3-4 for example. Is there any other downside apart from the obvious, that losing 2 seed phrases would lead you to losing all of your funds?
Finally, is there anything wrong if I used a private key both for a singlesig wallet and as a cosigner in a multisig wallet? I suppose this question must have already been asked somewhere in the forum