Yes, that blue box is a K40. K40 is just a broad name for a 40W CO2 laser with working area of about 200 mm x 300 mm. There are many manufacturers of it, and each makes a little bit different product. The prices vary a lot, and additional features are as varied as it gets.
My K40 laser itself is almost silent. There is some noise from the power supply fan, but not much. The loud parts are air evacuation fan (absolutely necessary) and the air assist pump (highly recommended). K40 is water cooled, so you'll need a water pump as well (they are not loud) and either a big bucket of water or a dedicated water chiller (which may or may not be loud depending on type, I don't have it so can't tell from experience).
If you're already done with the mechanical parts for the machine, then a CO2 laser tube such as ones found in K40 is probably not suitable for you. The laser tube itself is about half a meter long, so you likely won't be able to fit it on the machine. The usual approach with K40 is to have the laser tube fixed in the back of the machine and move mirrors that guide the laser beam where it needs to go, then finally use a lens to concentrate it just before the material you cut.
Cutting plastic is tricky. Acrylic (Polymethyl methacrylate, plexiglass, perspex) cuts perfectly on my laser, but stinks (remember air evacuation). That is pretty much the only plastic that is good for cutting with a laser. Styrene, which is otherwise excellent for hobby work does not cut well - it melts and deforms along the cut. PVC cuts ok-ish, but contains chlorine and during cutting releases it as chlorine gas, which forms hydrochloric acid with moisture in air. The result is that everything in the vicinity will rust very quickly, not to mention the health issues related to inhaling it. So keep that in mind, the only plastic you should cut with a laser is acrylic.
Wood (specifically plywood) cuts nicely. I had excellent results with 3mm ply, but again, not all plywood is the same. There are special ones designed for laser cutting and if you decide to go that route you should stick to those. Regular plywood doesn't cut cleanly, it just burns and leaves horrible edges. From what I gathered in my research into this, the problem is the type of glue used in regular plywood. I haven't tried cutting natural wood, mainly because I don't have any nice thin pieces, but I expect it to work fine.
In any case, you should have a strong ventilation to suck the air from the laser enclosure and dump it outside the building, or you could invest in a (rather expensive) air filtration system.
Moffy said it already, but it can never be said enough times, please have an enclosure around the machine. At these laser powers you can end up blind from a momentary exposure to the laser. A sudden unexpected reflection from something shiny is enough to permanently damage eyes. At these wavelengths eyes can't see the laser, and there is no blinking reflex to protect you. The second video you posted is exactly the thing you should not do. Having the laser beam in open space is dangerous (acrylic box is suitable, but metal is better if you can manage it). In addition to the enclosure, get yourself some safety glasses that are designed for laser wavelength you use.