I have 3 W's on my college transcript. Could it hurt my transfer status?

Yes, but maybe not by very much. This indicates a pattern of behavior where you are unable to handle the course load of classes or do not have the academic ability to pass several classes. That would call into questions your readiness to do the work at the next school. No university wants to enroll a student that is going to fail or drop out.

For transfers, the admissions staff will look at your GPA, the number of credits you have earned, and your intended major and assess you against all the other transfer applicants for that major. Any "red flags" will be considered. If you have a great GPA and most of the other transfer applicants have also dropped a few classes, your Ws won't be a big issue. If the school receives more than enough other applicants who have better GPAs and fewer Ws, it will be.

It could unless you explain them. Admissions is a holistic process- they'll look at everything. If it looks like you're a student who drops a class whenever you're not doing well, that could be a problem. If you can say that you had a health problem, for example, one semester and had to drop a couple of courses, they probably wouldn't hold it against you.