We save to our servers, so you need internet to play. If you close the game tab, then you get offline gold and experience points, and event currency if one is running. The auto-progress button, unlocked after your first reset, only works while the game is open to move your Crusaders through the areas until they reach one where the die.
If you press the pause button while it's already paused, the screen blurs more, and more pause menus pop up. But when you close the top menu, the game starts in the background while you have to close the following menus.
This line of games misses a couple very basic staples of the tower defense genre. First, you can't sell placed units. Second, you can't view the range of a unit after you've already placed them. If it had those two changes, that'd be worth at least another star.
Thanks for the suggestion. I've added it to our list to consider.