Is the `addmultisignature` API call dependent on order of public keys in the array? Or is it lexicographically sorted? Say Alice and Bob are part of a partnership business that accepts Bitcoin. They store all their main funds in cold storage until the end of 3 months, when they cash out and use a new address. This cold storage is in a single address.
Both Alice and Bob have full nodes, running Bitcoin Core. They use the console feature to generate a 2-of-2 P2SH multisig address, using addmultisignature.
They exchange each other's public keys, and input them as arguments.
If Alice used the following syntax:
addmultisignature 2 '[&lt;alice's key&gt;,&lt;bob's key&gt;]'
and Bob used:
addmultisignature 2 '[&lt;bob's key&gt;,&lt;alice's key&gt;]'
Will they end up with the script hash and address?
Follow up questions:
Can you give certain keys more &quot;weighting&quot; by specifying them more than once?
Does the same logic apply to SegWit (P2WSH) multisigs?
Is the addmultisignature API call dependent on order of public keys in the array? Or is it lexicographically sorted?

Yes. It is not sorted according to BIP67 automatically.
How did I confirm?
I used the below command in bitcoin core:

addmultisigaddress 2 "[\"0318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e0\",\"03e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c19\",\"02667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054\"]"

to get the multisig address details:

{
  "address": "tb1qlfu3fy8apmm4j73kzlxh9s09rtu6vzeru3l0asfd5ylwfqs7e0zs3hfdpt",
  "redeemScript": "52210318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e02103e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c192102667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b05453ae",
  "descriptor": "wsh(multi(2,[85ac4db2]0318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e0,[90060439]03e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c19,[98725f4c]02667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054))#qmw7suql"
}

Tried using the same public keys and the order to create a multisig address in caravan and got the warning that keys are not ordered.

Multisig address and Witness script returned are same for both. It changes when I order the keys in caravan:
Address:  tb1q6xnkaa3sej2x580yjc5j20pcy6x0ddxxc8a9ajlvvn2de2htvvpsnp95vd 
Witness Script: 522102667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054210318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e02103e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c1953ae