Suppose you're on the midpoint of a straight line 6e8 m long, which the dot traverses at uniform velocity.
If the dot moves at c, you would see nothing for 1s. Then there would be an instant where the first half of the line is entirely illuminated, as light from the dot at each point reaches you simultaneously.
After that, it would look like an ordinary dot moving away from you at c/2: if the dot moves for x seconds, it takes 2x seconds for the light from the dot at that point to reach you. (Note that a dot moving at c/2 would appear to be moving at c/3.)
If it's moving faster than light (say, kc, k>1), then when the dot arrives at you, you'd see a dot appear at your position, then move back to the original position at (k c / (k-1)), and vanish. (k/(k-1) is because you see it reach the start point 1s after it starts moving, and you first see it 1/k s after it starts moving.) If the dot had been stationary at that point before it began moving, you'd still see the dot there until the new one reached it, at which point they would both vanish.
Simultaneously, you'd see a dot move from your position to the end position at (k * c / (k+1)).
I'm not going to try to work out what happens if you're just standing near the path of the dot.
(I'm not sure about this, but it's what I get when I try to work it out. I'm especially not sure that I'm allowed to discount relativistic effects. I think I am because there are no massive bodies undergoing acceleration, but I don't pretend to fully understand relativity.)
I think philh already explained it pretty well. In your question you suppose the laser dot to be 'fairly large', which I think would complicate things (the shape of the dot 'smearing out' as it goes faster). For me, philh's explanation works best if you imagine the laser pointer to be a cannon of photons, emitting one photon after another at a very high rate and in a single direction, with the electrons of the moon's surface always reflecting the photons towards your eye (perhaps a bunch of strategically placed mirrors...).
Actually, the shape of the dot is unchanged by going faster. Since it isn't a real object, it isn't affected by relativistic contraction. If you emit a circle, it'll be a circle.
It is of course changed by what it gets reflected off of, but that's not special to this situation at all. If you're at the center of an enormous hollow sphere and shine an astronomically-bright laser circle around, it'll always be a circle to you no matter how fast you move it.
If the laser emits all photons constituting the dot at the same time, in pulses... I was thinking of the photons that make up the dot arriving at random intervals, but I'm probably making things too complicated.
