Questions on first ESA claim.

Hi. Just a quick question regarding ESA.

A Dr has advised me to claim ESA due to an ongoing illness. If I apply for ESA, how long would I have to wait for the initial payment (assuming one is made)?

I've read that the £70 per week for initial assessment is only for 13-weeks - is this set in stone, or can it take longer?

Finally, I need to see the Dr to get a sickness cert. Should I send off the application now, or wait a couple of weeks until I get the cert? I'd thought making a claim asap is paramount in getting the ball rolling.