Does the latest release of Armory need the Bitcoin Core Client to be configured with a "bitcoin.conf" file containing this line of code "txindex=1" inside the file?
no, none do
Or is it okay for the Bitcoin Core Client to just download the blocks without no need for re-indexing the headers which is what the "txindex=1" inside the "bitcoin.conf" file is designed to do?
I don't think that's what txindex=1 does, but it doesn't affect Armory if you use that cli argument with your Bitcoin client (Bitcoin uses txindex=1 to allow you to search the bitcoin blockchain for txids, blockchain.info and others basically got the idea for their service from the introduction of the txindex feature)