How to best clean the inlet vent screen on a microhood

Most things you can use (paper towels, cloth) will leave lint behind. What you want is microfibre cloth. The best one is the cloth intended for eyeglasses, but they are small, so it takes time cleaning with them. You can buy a microfibre cloth intended for cleaning windows. It is somewhat expensive (Eur 2.5) and you can't reuse it for windows because you don't want it contaminated with window cleaner. But it works much quicker.

As for cleaning liquid, I don't trust anything but water. Screens have anti-glare coating, and you never know how even the mildest detergent will affect it. I haven't had a visible residue even when using very hard water, but of course, distilled water or filtered water is a good precaution if you have it.

For the cleaning itself, you have to dry the screen, or you'll get smear marks. So a single damp cloth doesn't work well. You have two choices - either use two cloths, or a cloth and a spraying bottle. With two cloths, clean with the wet (not dripping wet) cloth, and immediately dry with the other one, before the water has evaporated. With the spray bottle (use a clean one intended for flowers, not an empty window-cleaner one because of residues), you can sparingly spray the screen, then dry it with the cloth. By sparingly, I mean that there shouldn't be a part of the screen which is misted more than once. Too much water, and you will not only have a too wet cloth, but you also risk it dripping into the frame and damaging the electronics inside.

Putting the stove in the corner like you propose also gives you some nice space to work with for routing the pipe up into the ceiling / roof area or out an exerior wall.

Seems to me that re-routing the vent to the previous riser location is a total non-starter. Many hoods vent up into the small cabinet above the hood and your re-routed vent pipe is going to make a mess of the upper cabinets if it has to route nine feet across your kitchen.