Well your speakers dont REQUIRE 35 watts, thats just the recommended. They can go off 5 watts if thats what was powering them. And your head unit won't be "tapped out." It gives equal power to each speaker.
As far as amping, I would really really really suggest a 4 channel amp to go along with your 600w. You should be able to pick one up for under $150. Wiring all your speakers off a 2 channel 600w will be a HUGE pain in the a$$ and will sound like sh!t. Trust me...even if you can't afford the 4 channel amp now, use your head unit until you can.