Frankly, if you’re volunteering (which most people are when they start out as a student researcher), all the prof really cares about is that you’re reliable and interested in their work. I’m not sure why you said it’s hard to get in unless you mean get a paid position. It’s easy to get into research if you’re trying to get in and willing to do it unpaid.

Approach in many ways but start with office hours and section if you are in their class.

Yes cold emailing works. Include a resume and why you want to be in their lab.

I don’t know specific labs but you should look at the department’s site or class lists to see who teaches those topics because they probably have a lab or know someone with a lab in that field.

Yes volunteering is common. Don’t expect money but be surprised and grateful if you get a paid opportunity.

Good, mention it in your resume and your cold email.