Applying for PIs and CPs at the same time will not hurt your chances. In fact, for me, it actually helped- the woman who did my CP interview was the exact same woman who did it for the PI a day later- and she was happy to hear from me again, because I had made a positive impression on her!
applying for too many PIs may hurt your chances- especially if they see you're not really qualified for all of them. I would apply for 4 or 5 at the most, and even that might be too many for some. I applied to 5.