The minimum width of trace you can reliably produce with Cirqoid is 0.2mm (8 mils). It's the same for the clearance (distance between neighboring traces).
Yes, these packages have been successfully used on the boards produced by Cirqoid. It's possible to use chips with pitch (distance between centers of the pins, usually denoted by e letter on IC drawing in the data sheet) down to 0.4mm.
Most probably no. Layouts using BGA chips usually rely on getting significant part of the pins to the other layer (or layers on multi-layer boards). This requires the use of very fine vias under the chip, which is usually difficult to achieve with PCB milling technology.
There's no machine vision in Cirqoid, thus components placement is semi-automatic. This means the machine moves the component to the position above it's place on the board. Then you need to assess whether the component aligns with its landing pads and adjust if necessary. This way, ultimately the accuracy of placement depends on your vision.
Neither of them. Cirqoid comes with a built-in vacuum pump. It also comes with a built-in air tank. All you need is a hand pump (like the one you use on your bicycle). On the other hand, if you already have compressed air available, it can also be put to use - this will save you the effort of manual pumping.
Cirqoid has dual-voltage power supply, so it works all over the world.
There are several approaches to this.
The most common and by far the least expensive is soldering a thin wire (or some TTH component's lead) between layers. Depending on the number of vias on your board it can be as entertaining as watching grass grow, but quite often it's the fastest way to get them done.
Another possibility is to press copper rivets (sometimes called eyelets). Rivets can be pressed sufficiently flat, thus allowing to put them under ICs (which is impossible with soldering approach). Though they need more space (i.e. outside diameter of the via needs to be larger) and tend to be less reliable.
There are other possibilities, including the actual electroplating process, but these tend to be fairly sophisticated and expensive.
Here's the photo of the tool that we find most suitable for this purpose. Pay attention to the fact that the jaws are smooth, as serrated jaws can damage the pin.
While Cirqoid is controlled by a GCode (that's an industry standard when it comes to controlling CNC machines) and can be controlled by any software capable of producing GCode, we recommend using cirQWizard (see http://cirqwizard.org/ for more information).
There's quite a variety of software packages serving this purpose. We ourselves use Eagle. The other packages that are popular among our users include KiCAD, DesignSpark, OrCAD, Sprint-Layout, Proteus, Easy-PC, Altium Designer. There're certainly others. As long as the software supports export to Gerber and Excellon files, it can be used to design PCBs to be milled on Cirqoid.