Let's say I have an existing BTC wallet, which is not multisig. Can that wallet be the cosigner of a multisig wallet?
There is no limitation that requires multisig addresses to be created using special multisig keys. A key from an old wallet can be used in conjunction with new keys from a different wallet to create a multisig address and its redeem script.
You could even create a multisig address using a few different brain wallets if you wanted to (definitely not recommended, but I'm trying to illustrate a point).
To create a multisig address, you only need to provide the public key for each key pair being used and the number of keys required to sign in order to spend. From there a redeem script is generated which is a trivial script that must resolve to true in order for funds to be spent.
There are some pitfalls to be aware of though...
Let's say you have three key pairs:
Key Pair A - Private Key A - Public Key A
Key Pair B - Private Key B - Public Key B
Key Pair C - Private Key C - Public Key C
A multisig address can be created by providing Public Key A, B & C; however if you create a multisig address by providing Public Key A, C & B it will generate a different redeem script and different multisig address. This can potentially cause trouble when trying to spend if you didn't save the redeem script and you need to re-create it to spend. So the order of the keys matters in order to re-create the same redeem script in the future (or you can just save the redeem script).