user62368: 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 '[<alice's key>,<bob's key>]'
and Bob used:
addmultisignature 2 '[<bob's key>,<alice's key>]'
Will they end up with the script hash and address?
Follow up questions:
Can you give certain keys more "weighting" by specifying them more than once?
Does the same logic apply to SegWit (P2WSH) multisigs?

user-1: 
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

62368: <p>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.</p>
<p>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 <code>addmultisignature</code>.</p>
<p>They exchange each other's public keys, and input them as arguments.</p>
<p>If Alice used the following syntax:</p>
<p><code>addmultisignature 2 '[&lt;alice's key&gt;,&lt;bob's key&gt;]'</code></p>
<p>and Bob used:</p>
<p><code>addmultisignature 2 '[&lt;bob's key&gt;,&lt;alice's key&gt;]'</code></p>
<p>Will they end up with the script hash and address?</p>
<p><em>Follow up questions:</em></p>
<p>Can you give certain keys more &quot;weighting&quot; by specifying them more than once?</p>
<p>Does the same logic apply to SegWit (P2WSH) multisigs?</p>
 -1: <blockquote>
<p>Is the <code>addmultisignature</code> API call dependent on order of public keys in the array? Or is it lexicographically sorted?</p>
</blockquote>
<p>Yes. It is not sorted according to <a href="https://github.com/bitcoin/bips/blob/master/bip-0067.mediawiki" rel="nofollow noreferrer">BIP67</a> automatically.</p>
<p>How did I confirm?</p>
<p>I used the below command in bitcoin core:</p>
<pre><code>
addmultisigaddress 2 "[\"0318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e0\",\"03e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c19\",\"02667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054\"]"
</pre></code>
<p>to get the multisig address details:</p>
<pre><code>
{
  "address": "tb1qlfu3fy8apmm4j73kzlxh9s09rtu6vzeru3l0asfd5ylwfqs7e0zs3hfdpt",
  "redeemScript": "52210318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e02103e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c192102667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b05453ae",
  "descriptor": "wsh(multi(2,[85ac4db2]0318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e0,[90060439]03e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c19,[98725f4c]02667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054))#qmw7suql"
}
</pre></code>
<p>Tried using the same public keys and the order to create a multisig address in <a href="https://unchained-capital.github.io/caravan" rel="nofollow noreferrer">caravan</a> and got the warning that keys are not ordered.</p>
<p><a href="https://i.stack.imgur.com/zj60m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zj60m.png" alt="caravan-multisig" /></a></p>
<p>Multisig address and Witness script returned are same for both. It changes when I order the keys in caravan:</p>
<p>Address: <pre><code> tb1q6xnkaa3sej2x580yjc5j20pcy6x0ddxxc8a9ajlvvn2de2htvvpsnp95vd </pre></code></p>
<p>Witness Script: <pre><code>522102667b6e09a4a82b12cc8e5611373f28673f2cdd2e42b2f7e89825ec82b7f9b054210318370919cfceb3d260081eeca3cae19f941eec321f597c48a64839178ea1e3e02103e783b8cf0474f732cc5d89cc893c25ab8a487785eb541a3ba0b26cfdd38c5c1953ae</pre></code></p>
