- Electric fields point away from positive charge and towards negative charges.
Don't think about the cylinder drawn. The only think that matters for the field direction is the charge in the wire.
Think of it like this: The electric field at any point points in the direction a positive charge would move. That means, a positive charge would move away from another positive charge (therefore the field arrows in the drawing you have point away from the positive charged wire in your figure.)
- The $dA$ parts are simply small areas. If I were to talk about a small area of the cylinder surface, then I would also like to know which side is inwards and which side is outwards in the cylinder in the figure. Therefore the areas are given a direction at the same time. This is called a normal to the area.
Then if you couldn't see the cylinder, but I told you that I have an area $d \vec A_1$ that points to the left, you would know that the cylinder is on the right hand side of this portion of the surface area. It is simply a matter of definition.
Area $d \vec A_1$ points left because it is on the left end of the cylinder (so it points outwards from the cylinder).
Area $d \vec A_2$ points right because it is on the right end of the cylinder.
Area $d \vec A_3$ is placed at the curved cylinder surface. It therefore points out from the surface away from the cylinder.
dA2 and dA3 are in the same direction of the E3 and E2.
$d\vec A_2$ and $E_2$ are not in the same direction on the drawing. I don't know what you mean here.
$d\vec A_3$ and $E_3$ are indeed in the same direction on the drawing. This is simply because, the $d\vec A_3$ will point straight out of the curved cylinder surface (perpendicular to the surface) at any point on the curved surface - and $E_3$ happens to do exactly the same, since it always points away from the wire in the center of the cylinder. So at any point on that curved surface, those two point in the same direction.