Before intervention, it's velocity points perpendicular to the direction of the laser fire.
Maybe you can grok it by thinking of the laser as excerting torque on its current path, twisting it so that it soars higher initially but afterwards comes closer to the center of gravity of earth?
No, you've deflected it in a direction perpendicular to the one it's moving in; turning it slightly away from the Earth, initially, but not speeding it up. This will increase the eccentricity of the orbit.
It depends on the angle at which you hit the debris with the laser, as well as the shape of the debris.
The idea is, generally, to hit the debris at an angle as it approaches you (bottom front of you may)[1]. Depending on the shape of the object as well as the material of the surface, you'll be able to transfer enough momentum to actually slow it down as well as nudge it further (loss of mass?) out into a new orbit with a low enough perigee.
It's obviously not easy to predict the new orbit because debris comes in very different shapes, and of many different materials.
Also, because there's a transfer of energy, the part of its original Vx is lost in the process (the laser is deflected at an angle). There's no loss of total Vx (sum of laser Vx and debris Vx) however.
That seems unlikely, but even in that case you'll achieve a lower perigee. It's also extremely unlikely that it would happen - hitting debris when it's directly overhead, yes, but hitting it on a surface that's perpendicular to the laser, like a letter T, probably not.
But assuming we do hit it like a T, the two things happen: loss of mass (laser ablation) and transfer of momentum. Vy will increase but Vx remains constant. The debris is moving in a straight line - that speed doesn't increase, but now it's moving at a different orbit so, relative to its previous orbit, Vx will appear to have reduced (but it hasn't).
But I'm definitely not an expert in orbital mechanics so don't quote me on that.
Okay, you're right, it does speed up very very slightly, but by an amount that is not anywhere near sufficient to keep the perigee from falling, as you originally claimed. The increase in eccentricity of the orbit is the larger effect.