yes can. No problem at all but make sure to have re entry permit before leaving Thailand. If you leave thailand without this then your visa will be finish
The requirements may be referred to immigration to immigration but in general they need those two documents you mentioned plus rent contract. I'm sure with this immigration will issue residence proof
If you enter Thailand on 60 days visa exemption and stay only few days and didn't make any extension then after going to Laos and reentering Thailand there will be no problem for you
There will be no problem for you as your traveling behavior is of genuine tourist. Also you didn't have any extension on any of your trip. There should be no problem. But yes prepared to answer a few questions from the immigration officials
please count from the latest (last) date you entered Thailand. From there if your continuous stay is 90 days or above then 90 days reporting is required. If your continuous stay is not 90 days and you plan to leave thailand then 90 days reporting is not required.