Assuming you mean immigration at an airport then all you need is your passport and last boarding card.
Check they give you the correct stamp while you are at the desk. Much easier for the IO to fix a mistake then than trying to get it rectified afterwards.
It’s completely different depending on the office you use. In Phuket as an example you need to have a recent filing if you’ve entered the country, stayed in a hotel recently or even a night in a hospital. Other offices don’t seem to have the same restrictions and although a recent stay somewhere may have “overwritten” your main residence on their system they have a history online so can verify your main address.
That should be fine. Where the agent thing is an issue is where they get an agent to do the extension without the correct funds. Usually it’s done in a province where you don’t live and possibly have never been.
There isn’t a limit as such. Each embassy or consulate will have its own rules. For example Penang in Malaysia will only issue one per year, but it you apply for another in KL you will probably get it.
In your home country they will probably issue without question.