There is no reason not to fire a Burst I and Burst II at exactly the same time. A second delay between them is probably not enough to allow a full shield layer to regenerate, but that's a risk you can run by stringing out your shots.

The only reason not to fire your weapons simultaneously is to sequence a particular weapon to fire last- i.e., if you have a Heavy Laser that deals 2 damage per shot, you want that shot to hit last after the 1-damage shots.

Depending on the enemy's Engines size and crew, it could have Evasion of up to 35% (I think that's as high as it gets, except for the Flagship). At that level of Evasion, it should usually dodge 1 out of 5 shots, and frequently dodge 2 out of 5.

You should eventually get hits in, but it will be slow-going. I don't think it's too improbable to go a full minute without hitting.

In a situation like this, you might want to try damaging Drone Control first to try to knock the Defense Drone 1 offline, then you can use your missile.

5 shots is light on offense when you are facing ships with 3+ layers of Shields. You'd ideally want to add another weapon, or add systems that give you offense or counter enemy defenses.