Activating Twinning Glass's ability allows you to cast a single card from your hand as it resolves. It doesn't create a continuous effect, and it doesn't let you cast multiple cards for free.

It doesn't matter who cast the first spell.

You can't pay any alternative costs (such as those from evoke or morph) when casting the card. On the other hand, if the card has additional costs (such as those from kicker or buyback), you may pay those.