First, should I wait for this PI to tell me if I can return or not, then from there look for a new lab? I want to continue to do research in some way because it really interests me, but that means that I should start emailing professors today if I can't return to my old lab. Is it okay to start emailing PIs without a final "yes" or "no" from him?
You need to make clean relationships. Decide if you want to return to the lab or not before emailing other PIs. Commitment to a single research project is key to meaningful research. It looks bad if you're jumping around without making your intentions clear with other people. You may explain to your first PI that you would like to explore other research topics before emailing other PIs. Also think about if you will list your first research experience when emailing other PIs. They will surely email/call your first PI if you do.
Second, I plan on meeting with him this week to discuss my grade, and hopefully explain myself in some situations and get a higher grade. Not try to beg him or anything, and if it turns out he does not want to change my grade, I should leave it here. But I am not sure. Should I try to be a little aggresive and get a higher grade if it might help? I have done that for some of my classes, but I am not sure about this situation. This might affect what I have below, though.
DO NOT ASK FOR A HIGHER GRADE. This is in incredibly poor taste. You recognize the poor job you did and acknowledge you deserve that grade. Faculty know one another, and again, if you want to list your first research experience when looking for other research opportunities, they will contact your first PI.
Third, I really wanted to do some summer research programs, and I was hoping to ask my PI for a LOR. I am not sure I should anymore due to the grade he gave me and his hesitancy to let me back into the lab. That being said, he does know I am someone who cares about the work I do and I have always been respectful in the lab to him. Maybe I should just stick to professors for LORs, butI am not sure how much that will help.
When asking for a LOR, confirm that he will write a strong LOR supporting your application, in those exact words. I think that longitudinal research experiences (i.e. way more than a semester of a summer, ideally one experience throughout college) is the most valuable. I prefer 2-3 year commitments to a project at the home institution than a summer away, though summer research programs are a nice change of pace and exposure.
Finally, if I do go on to look for another lab, I was thinking I might ask him for atleast a recommendation on my CV. Like I said, I didn't have great work, but I was always respectful and open to learn. Should I ask him for this recommendation when I meet with him this week?
If you want to list an experience on your CV, it's always good to give the listed contact a heads up. If he feels uncomfortable supporting you should somebody contact him, he should advise you to keep this experience off your CV.
As you can see, I am pretty conflicted about this and have a good amount of anxiety from it. Any advice would be greatly appreciated!
Thanks so much.