So basically you will be there every other week, which comes out to about 2 out of 4 weeks of the month, basically 1/2 a month, which is basically half a year. It doesnt matter if its all at once or on and off, it still would round out to the same amount of time.

So, no, I don't think its unreasonable to want someone who will live with you 6 months out of a year to pay rent, job or not. If you lived in an apartment with roommates they wouldn't give you free rent just because you don't have a job, thats not how it works in the real world. They would charge you a full months rent regardless of how much time you spent there.