Q: Time Train and It’s Your Lucky Day both say to play an extra card after. Is this mandatory?

A: It is never mandatory to play a card. You may choose to draw instead and end your turn (but you must do one or the other). As the rules say “You can exercise this option at any time you are called upon to play a card, such as when you get an extra play because you used a really fancy time machine,” (like the Time Train).