How are components with contacts under the ICs soldered onto boards?
This is a ball-grid array (BGA) package. Each little blob is actually solder. The component is placed precisely by a pick-and-place machine, and then the whole board is heated up to melt the solder (reflow soldering). The solder will flow to the pads on the PCB by surface tension, and the solder mask (hopefully!) stops it shorting out nearby pins. The advantage of the BGA package is the much greater pin density (and hence pin count) available compared to inline or QFN packages. You will typically find high pin count (FPGAs etc) components only available as BGA as it's impossible to bring out 1000 pins to the edges.

These are difficult to work with for prototyping work as it's nearly impossible to tell if the solder joint has been made. In mass production, this is done by X-ray imaging. You can make a reflow oven at home, but as you say the precise placement is quite tricky by hand, although not impossible.

Reworking is very difficult, you generally have to remove the whole component and reapply the solder blobs to remake it. Typically, vendors offer development boards with the BGA chip presoldered with the pins brought out to header pins, which are much easier to deal with.