you can move freely between T1 and T3, but of course you have to go through security after arrival (even with a T3/T3 connection), as is the case after most (except intra-Schengen and some others...) international flights at any airport in the world.
i have found that the queues for security between T1 and T3 can be a problem. It once took me more than 30 minutes queuing and that was in Business - worse was the fact that it turned out to be quicker in the economy lanes
I have not used Etihad for a while but will be interested to see if things have improved