Use Relic charges immediately when you get to lane at level 1. The first charge will refresh in time to get the first siege minion, and doing this helps rush level 2 for the engage. And be especially sure that you get all siege minions.
After landing your E, sometimes it is better to AA once before hitting Q, especially when Morg has not taken her black shield yet. The one extra auto only takes a fraction of a second and could have brought Cait low enough for Ez to finish her. The other option is to E in but save your Q in case Morg shields. Just AA and walk until the shield wears off (if you think a long trade is in your favor) or use the Q on whoever didn't get shielded.
Ez was totally correct to push before backing. The reason he was already losing minions to the tower when he returned to lane wasn't because he pushed before he backed; it was because he spent 25 seconds shopping. Under normal circumstances, he would have made it back in time to miss fewer minions than Cait did.
At 4:10, you're standing in the bush with traps. You don't know if it's warded, but you know they know you're in that bush. That's no secret. And you know Morg is in the opposite bush, and her binding is up. That wasn't a good time to be idling in a bush in the first place because it's easy for Morg to land her snare even without vision, but then you made it worse by walking onto the visible trap and needing to flash out after back-to-back snares.
Right after that, you start recalling with 750 gold. I would definitely hold out for the 800 for Sightstone. Fortunately Morg interrupts you and you do have 800 gold when you recall, but you don't get a Sightstone! It is almost always worth rushing as your first item. Vision in the lane bushes will help avoid those surprise bindings and you'll be able to ward the river too. Instead you pick up Targon's, return to lane, and ignore the huuuge wave (with a siege minion) that you could have used all 3 charges on.
Hey, if you wanna swing by the stream, I should be on tonight and tomorrow night to do a full spectate for you if you're interested. I have you on my friends list but pretty sure I did not get to spectate you yet. Stream link is here.