No feat tax. Withdrawing or stowing a weapon is a 'free' action. But with the feat it's not needed, but you gain the ability to cast as opportunity attack and advantage to concentration. Casting isn't necessary to make an eldritch actually super good at combat, but it offers him the versatility...
False. Feat is called Warcaster, and was made for fighter/caster types (fighters get a few feats btw ;)). Also not only that, he can eventually cast and melee attack in the same turn! And even moreover Eldritch knight probably is the 'toughest' and likely has the best multi-encounter DPR of the...