When user 'A' creates a transaction using scriptpubkey, do they also sign a message with their private key? I understand a transaction in Bitcoin is created in two parts:

1) Sender 'A' creates a claim condition using scriptPubKey

2) Recipient 'B' proves ownership of the recipient address using scriptSig

I am wondering whether step 1 also involves the sender 'A' signing something as a part of the claim condition using their own private key?
No, the scriptPubKey doesn't require 'A' to sign anything. It only asserts what B must do to spend the output.