But if you do have any convictions then you wouldn't be applying for the ESTA anyway as you would need a visa from the US embassy.
Taken from the U.S Department of Homeland Security site.
I have a visa? No. ESTA is not required for people traveling to the U.S. with a valid visa. However, if you have a visa, but are traveling under the visa waiver program (VWP) instead of your visa, then you will need to apply for ESTA.
You can apply for the ESTA anytime before travel but they advise at least 72 hours before and, as said before, it last for 2 years.
Not at all. If you only plan on going once in the next 2 years then the earlier you do it the better - but if you are planning any other trips than I would leave it until nearer the time so it lasts for longer.