Can an unsynchronized node create a valid transaction?
Node A is holding a chain which is not in sync with the rest of the nodes and node A decides to post a transaction.


Your node, that is not synced, will only let you spend outputs that it has validated as unspent. So in your scenario step 1 may not be able to happen depending on age of the UTXO and what block your node is at. An old node that is not synced can only spend old coins that haven't moved. 

A more detailed answer from A. Chow: https://bitcoin.stackexchange.com/a/91822/26873