Yes, you have the choice to stay with your original choice or to switch.
No. Think of it this way: If you picked one of the two wrong doors at the beginning, you'll always get to the correct door if you switch (because the second wrong door will have been eliminated). If you picked the right door at the beginning, you'll get a wrong door if you switch. So what is the chance that you win if you switch? 2/3 because that's the chance of picking a wrong door at the beginning.