I would definitely recommend getting the IDP from home before traveling to Thailand. It gives you some breathing room and many DLT offices do like to see it.
You don't need a lease nor any minimum stay. You'll need a TM30 though. You'll also need to be staying at that address long enough for them to mail the certificate to you as that's how they do it to ensure you're actually staying at that address. So it can take 1-2 weeks after you apply for it to arrive. You'll need to be able to receive mail there.
Some banks might require a lease along with the residence certificate and some might not.
I don't think they even check for them anymore. Doubt it will cause you any issue, especially since you don't even see a passport control officer anymore. You just scan your passport at automated exit gates.
I really have no idea what you're talking about. Of course it shows up. I'm not sure what gave you the idea that it doesn't. It's a regular transfer in and out.
That would be a question for your local immigration office. Not every office allows conversions to non-O. For the offices that do, they each have their own requirements. You should go there and ask if they allow it and for their printed list of requirements. The primary thing is going to be proof of 800,000 baht that's in your Thai bank account for 2 full months already and that you can show was transferred from overseas.