If you apply from your home country, use a utility bill, bank statement or such (as others already suggested).
If you apply from another country, i.e. another country in SEA, then take a picture of the entry stamp in your passport and merge it with a hotel booking/reciept.
I think one can assume that they run some background checks. From what I understand, Thailands future plan to use an electronic travel authorization for visa exempt travellers will, if and when being launched, involve security screening of foreigners entering Thailand... that is the plan for people arriving without a visa and I can't imagine them not doing the same for ANY visa applicants... but as I wrote, this is just me guessing.
I highly doubt that they will ask DTV-applicants to submit police records from their home countries (unless you are a national from a country with special restrictions to obtain a visa, i.e. Nigeria).
In your home country or in a country that has a Thai embassy that accept applications from non-residents. Some embassies have restrictions for nationals from specific countries, so without knowing your nationality it is impossible to answer.
This question is asked every day - next time, use the search function...
No, you are supposed to be in the country you apply and also to wait until the visa is approved.
Sure, some people have gone back to Thailand and then exited and re-entered to activate their DTV, but you are not supposed to...
Why? Well for several reasons - two being that the embassies reserves the right to call you in for an interview if they see a need to do so AND there's been several reports of requests for additional documentation such as pics of all pages in the applicants passport (to confirm they are still in the country).
So just stay! What's the rush? - it's a 5 year visa.
You can keep your draft and edit your application once you get there (not just hit the send button). As proof of current location you should add the entry stamp in your passport together with hotel address/reciept, so you can't really complete until you get there. There in no rule that you can't add data to the application before you arrive to the country where you apply.
It's only been 2 working days since they started applications through the E-visa platform... It's unlikely that anyone can share any information about processing time yet.