Is the `addmultisignature` API call dependent on order of public keys in the array? Or is it lexicographically sorted?
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