How to properly use paid participation survey in my research?
I'm not very familiar with paid participation surveys, but anyway the first step is always to seek approval by the ethics committee in your institution. They might require  modifications in your methodology if they find some issues, and their approval will ensure that your study complies with minimal ethical standards.

In a publication you should definitely mention that participants were paid, since this is an important information about how the data was obtained. In general you should always explain the details about the recruitment method. Typically it could be a paragraph at the beginning of a "data" section, I'm not aware of any strict standard about this.

I've seen many cases where participants are offered a gift voucher of some kind, so I'd say it's quite common to compensate participants. There are also many studies which use crowdsourcing platforms where participants are paid for their time. I assume that it's also common in clinical trials but I'm not in the medical domain myself.