1. It depends. Do you expect the property you are moving out cash flow? What phase is the city you live in the real estate cycle? Do you feel comfortable managing the property? If it cash flows and you are comfortable with renting, i would say rent it. If you are in a seller's market, maybe cash out and hold the cash for other investments or buy discounted properties when it becomes a buyer cycle. You might consider doing a 1031 real estate exchange and roll the profits from first home into next one tax free.
2. Ethically, whoever buys the house has to live there. Also, beware that some banks are particular about what property you buy next if you already own one. For example, if you own a single family and want to buy a four plex with FHA, some banks would no allow that. Going from let's say a single to another single is probably ok. I ran into this problem this year.
3. I recommend asking the bank whether this would be cause for concern. Some banks are ok with it.