Lots of people enclose their computers to keep them quiet. It's certainly do-able, as long as you take care to have adequate ventilation to avoid heat buildup. (Although I would expect overheating to be less of a problem in Sweden than, say, Texas.) I'd try to find a way to place some vent holes above the computer.
As tlw notes, openings do somewhat defeat sound isolation. That, however, can be easily mitigated by thoughtful placement of those openings. They need not be immediately above the computer, as heat will find its way out even if the exit path is circuitous. In fact, circuitous airflow is a standard method for reducing sound transmission in heating and air-conditioning systems.
As for cable openings, just wrap some insulation around them. The frequencies you're trying to dampen are easily absorbed by fiberglass insulation. Consider laying in some rigid fiberglass along the inside walls of the closet.
USB cable length might be your biggest challenge. In my experience, long runs can be unreliable, even at the official maximum of 5m/15ft. Invest in a very high-quality cable, obviously. But if there is any way to get the interface closer, that would be best. For me it meant moving my whole rack to the opposite side of the desk - a major undertaking but it allowed me to use a shorter cable.
If that's not possible, you can use a USB repeater. There are three types.
This one uses Cat5 cable. It will introduce a very slight bit of latency, but not enough to be concerned about. There are also active cables and repeater boxes with passive cables.
Here's some reading material (pdf) that might help.