The favor that Alicent gives to Daemon is clearly seen on his lance when handed to him after the announcement he is to face Criston Cole. It is then missing when their horses rearing up just after this, but is back again during the joust and can be seen flying up the pole when the two men clash.
When Viserys is talking to Aemma in the bath she says to him "Nothing will cause this baby to grow a cock if it does not possess one." The word cock used in this sense was not commonly used until the 17th century, over 1,500 years after House of the Dragon was set. This series is not set 1,500 years prior to the 17th century and indeed is not even set on Earth at all; rather it is set in its own self contained fictional universe. There is therefore no anachronism in the use of this term.