How do I approach a professor for a self-designed PhD in Psychology? (International student, finishing Masters)

Hi everyone! I'm finishing my Masters in Psychology and want to pursue a self-designed PhD abroad, mainly looking at Europe. I want to cold email professors whose research aligns with mine, but I'm not sure how to go about it. A few questions:

1- How do I find the right professors to contact?

2- What should a cold email include, and how long should it be?

3- Should I mention I'm self-funded upfront or later?

My background is qualitative research. Any advice would be really appreciated! 🙏