The purpose of the "band" you identify with the red arrow that doesn't have any copper solder pads, is likely for 0.300 inch pitch DIP sockets and ICs; that is, the part straddles the gap, just like on a regular prototyping breadboard, where the part straddles a valley on the board. Yes, you can cut the board, but be aware that PCB material is pretty abrasive stuff, and may dull blades over time. Also, were eye protection and a face mask (ideally a respirator); the fiberglass and other components of the board (as dust from cutting) are hazardous to breath and will irritate the eyes. To clean such a board for soldering, light detergent, a green plastic scrubby, and a final rinse with distilled water then thoroughly drying it will work fine. You don't show the other side of the board, so I don't know if the board has copper pads on the other side; if it doesn't, then you can't solder to that side. If it does, then it is best to solder the parts on both sides where you can (unless the holes have full thru-plated holes); it will produce a stronger joint (some parts, like DIP sockets and headers, you probably can't do this). Finally, as far as joining rows and such, you can purchase special "wire" for this (called "tinned bus wire" - which is pre-tinned with solder and has no insulation) - or you can save component leads as you cut them. Some hobbyists like to bend the leads of components where they can and solder the circuit together using the leads (tacking them down to intervening pads along the way). You could also strip the insulation from some thin solid copper wire (like inside most cat5e cable), and solder that on (run solder over the wire to "tin" it and prevent corrosion over time).