The "conditional acceptance" is very much just that. If you go to Mills on a two year program, after you finish the first year, you can apply to Tulane. You usually interview by December and find out by March, when you're given a tentative acceptance based on scoring an acceptable score on the MCAT (don't know the exact number). If you do okay, you start med school in the fall. Essentially, it's a way to avoid the glide year. Only so many students at Mills can apply via this program (they're selected by the administration). I don't know how competitive it is. I don't know anyone doing it.
The linkage with Western Osteopathic works the same, except it can be done by those on a one year postbac with Mills. I don't think that there's a limit to the number of students applying.
Mills is a good program. Very high level of classes and very good students you're working with. It really is very cooperative. A's can be gotten, but they're not easy. It's a very well respected program. I like it.