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No, they mean that all applicants must have years of work experience. Some applicants might be considered if they never got a college degree, but they prefer applicants who got a college degree before they got their six years of job experience.

Ah that makes sense. Graduates sounds to me like people who just graduated, but of course that's not what it technically means.

There's no contradiction; it's just not an entry-level job. (Someone could get work experience then a degree, or a degree then work experience, or both simultaneously.)

Your quotation doesn't says work experience. It says experiences with [some computer] -- you could get that at a university.

Seems odd though, two to six years. If you used it in a few classes during 3 years, I'm not sure I'd even call that 3 years of work experience. And asking for six years is really weird in this context.