The purpose of a fuse is to prevent a wiring fire or tracks burning off a PCB if a semiconductor fails or some other short-circuit occurs. In general, semiconductors blow before fuses do. Don't expect a fuse to prevent damage to your electronics, although the odds of the board being repairable are much higher if the circuit is appropriately fused.
Unless you are using excessively thin hookup wire that cant handle 10A, or the PCB has seriously undersized tracks, you probably only need one fuse as close to the battery as possible.
Re: soldering wires to a LM2596 module or similar. Yes, that's how its usually done, unless you use pins for a direct board to board mounting, but soldered stranded wires without strain relief are unsatisfactory in high vibration environments and tend to fail by fatigue where the individual strands enter the solder joint . e.g. if you are mounting it in a RC helicopter you'd probably need to crimp terminals on the wires intended for direct PCB mounting, that also grip the wire insulation and solder the terminals in the holes to prevent any solder wicking up the wires.
Caution: many LM2596 modules cant deliver their rated current. Don't expect more than 2A without additional cooling, and if its got a fake LM2596 chip, you'll be lucky if it can do much over 1A. Fakes that operate at ~50KHz instead of the 150KHz of the genuine chip are extremely common, with the problem being that the inductor and capacitors aren't big enough for the lower frequency.