Can you install a window air conditioner through a wall?

Not necessarily. You want to make sure that the window unit has a slide out chassis. Most small window units are not. The problem with air conditioners for the window where you can't slide the guts out of the sleeve means that they usually have ventilation fins on the side which could be blocked by the wall depth. If the wall blocks part of the fins, then moisture is allowed inside the wall, which could lead to problems down the line. The main problem with blocking the side fins, are that the unit won't breath correctly. This will lead to a less efficient machine in turns of BTU output of cooling and will consume more energy. The parts inside can overheat and can potentially be a fire hazard. At the very least, the unit will die prematurely. Besides that, also take into consideration that window units installed in the wall are not installed to "manufacturer's specifications" and the warranty will probably be void if the manufacturer finds out about it's installation. Finally, if the unit should require maintenance or service, you can't easily remove the unit without leaving a gaping hole in the wall. If you have a slide out chassis unit, the sleeve can be left in the wall and capped while the chassis/guts are being serviced.