Include the position as "Incoming Summer Associate", or if in doubt contact graduate recruiter working for the company you're applying to
If you were accepted in competitive summer internship program, that's something that will be important for screeners/interviewers, even if you haven't started/finished it yet. I work as a software engineer, and if I was put in similar position (that is recruiting someone in April for an internship next summer), I would want to know if this person was accepted into GAFA internship this summer. If they passed Google interview process, they're good and I should talk to them.
Caveat I can see: I'm not familiar how law firms structure their internships program (1), but you may be asked why you don't want to continue your career with the firm you have internship this summer, or what you will do if you have an offer after this year internship. Internship programs are mostly designed as an employer branding tool that enables companies to recruit more efficiently for full-time graduate role, so you want to show during the interview that you will stick with them longer if possible.
Lastly a lot of companies organize networking events, or (in current situation) networking webinars, when you have possibility to ask questions. Ask recruiter directly if they want to see what you'll be doing this summer! If no events available, try to approach graduate recruiters through LinkedIn or email
(1) E.g. investment banks often has spring programs for students who have two more year till graduation and summer programs for students who have one more year till graduation, with assumption that good interns from spring program will join summer program, and good interns from summer program will join as graduate workers
I haven't answered question in full:
How would you recommend listing the dates worked?
If you know start/end date, you can put it there. In other case you can just put "Summer 2020"
How would you recommend filling in the details of the role
You shouldn't put too many details in my opinion. You can mention specific project you're assigned to, or briefly describe how the program is structured (e.g. duration, team or teams you'll be working with, acceptance rate if known)