How did you do with your TEAS?
Rasmussen only looks at your overall score, not your individual scores (math, science, etc). I believe you need an overall score of 55% or 65% to be eligible for their RN program, so even if you don't do so well on the math or science portion, if you do really well in the reading and english sections you should be OK.
As far as accreditation goes, they aren't accredited by the NLNAC or the SACS, so if you wanted to transfer out to USF or SPC, you more than likely won't be able to since Rasmussen is accredited by a different agency. However, you can transfer in your credits and apply them towards their RN program. This is a little off putting, especially if you want your BSN, however, they do have a BSN program which can be completed online.
Will attending any of these schools for RN program affect you in getting a job? No. I had the same concerns, and I personally called the nursing recruiter at Baycare to ask them if they gave preference to one college over the other. I was told, and this is verbatim, that they cannot hire someone based upon the school which they attended since they cannot show any bias from one institution to the other. Will they more than likely hire someone from USF or SPC? Maybe, but that also depends on the applicant's background (i.e do they have any experience in the medical field such as being a CNA, etc), and are they applying from within the organization? I asked if they have hired graduates from Rasmussen, and the nursing recruiter said that they have indeed hired students from Rasmussen, so yes, as long as you have a valid RN license from the state of Florida and a two year degree, you can be hired as an RN.
I hope I was able to answer some of your questions, if you want to know anything else, just let me know!