I just played a bit with CNC engraving and laser projection, so don't have that extensive knowledge, but AFAIK:
Even with your example of 10W, 10mm beam it has a bit more than 0.12 W/mm^2. Hundred times the power output from the sun. Even if we use a 1W laser, which should be enough for a mosquito, it's still like looking into the sun with a 10x magnifier.
And as you said - you focus the spot in 0.1mm with a small focal length - that's great for cutting or engraving but won't deliver enough energy to a mosquito 5 meters away unless you start with the same beam size, then focus it 5 meters away at the target. Only then it will need another 5 meters for the beam to return to the initial diameter and many more times that to finally be safe.
What I propose is a big reflector and aiming optics directed at it(a bit like a satellite antenna). I didn't do the math, but tens of centimetres or even more than a meter of dimeter. Let's say 50cm==500mm.
Then if you want to focus it to 5mm at 5 meters it will have less than 10mm at +-5cm. You could do the same with a laser but you just don't need to. Instead of 1W laser you can use a 10W powerled, have the same energy at the target but it will be much safer at almost every stage before it.
And what most people think about when they talk about shooting mosquitoes with a laser is, I think, sending a coherent 5mm beam directly - e.g. from a laser pointer. So you don't have to estimate the distance, but it is even more dangerous.