And what happens if arduino passes setup with ethernet.begin successfully, but after in loop state router goes offline then online - arduino needs to ethernet.begin again?
Not on my MikroTik routers. They keep dhcp lists in NVRAM, so outages don't affect them. Then the Arduino will renew when it is time if you are calling Ethernet.maintain() to renew the dhcp lease.
I'm checking with IDE v1.0.4 now, and Ethernet.maintain() function has a bit of a glitch. I haven't had a chance to pinpoint the problem, but it doesn't renew correctly on the first pass (at one half the expires_after). It "checks in" (last_seen is updated), but doesn't reset the expires_after time. Once the lease expires, it works ok, except it issues another ip address and marks the old ip as "busy".