New rounds start at about 1:00AM EST each day.
If you go to play and you have lost your home planet, the easiest thing to do is just create a new account. Many people who play have two or three so this isn't a big deal.
Once you get familiar with the game a bit more, you can write a script that will protect your home planet by mining it automatically and building up a small fleet around it.
I hope that helps! Let me know if you have nay more questions.
-Abstrct