"work experience" is correct. But does taking a gap year necessarily give a person valuable "work experience"? Some people would say a gap year gives a person valuable life experience because many do lower level jobs just to get away from the rigors and discipline of school. Just a thought.

The meaning can change, and in this context, you need to use "work experience". This means you have an understanding of an employment situation. "Working experience" suggests functional or practical experience, but not necessarily experience in employment for the first time. You can compare this phrase to "working knowledge".

'work experience' is more standard. Yes, the grammar is fine.

I would use 'work experience'. Maybe if you talked more about the jobs that you plan to do during the gap year then you can probably use 'working experience'. Doesn't matter that much to be honest.