The type and rating of the fuse you must use depends on your local regulations and laws, so it is difficult to help without knowing.
However, the regulations, no matter in which country, depend on physics - more precisely, they ask if the fuse will prevent the wires, switches and outlets from burning in case something fails and causes an overcurrent.
Despite your nice diagram, we can't tell what types (diameters/cross section) of wire are used in your installation, and where the wires are put. For example, different ratings apply depending on whether your wires are laid out in a stone wall or in a wooden construction, or in a wooden construction containing some thermal insulation making it harder for your wires to get rid of heat.
A typical installation in Europe, for example uses wires with a cross section of 1.5 mm2 and fuses with 16 A. If the cross section of the wires is smaller, the fuses have a lower rating, too. If the wires are put inside of thermal insulation material (e.g. in an outside wall or along the roof), you may have to use 13 A or even 10 A for the same 1.5 mm2 wire.
If you are not absolutely certain about your local regulations, please ask a professional electrician. If not done right, you put your house in danger of burning down.
Also, if you have sparks in your fuse holders, I think it's time to replace the entire fuse-and-holder combination. Sparks / arcing may cause very much heat, starting a fire in your fuse compartment!
The inductive kick can't increase the current. It will tend to maintain the existing current and thereby possibly cause high voltage someplace, but it doesn't ever increase the current.
In this case it will keep the current flowing a little longer than it would if there were no inductance. The 50 kA will melt or vaporise the fuse element, which eventually opens the circuit. The inductance will force the current to change more slowly, which causes the arc accross the dead fuse to be sustained a little longer. Most of the voltage will be accross the fuse at that point, which puts a reverse voltage on the inductance, which allows the current to ramp down to 0.
The energy stored in the inductance will mostly be dissipated in the arc as the fuse opens. Fuses are designed for some amount of inductance and should be able to take it.
Best Answer
Little fuse has this app note. It's very good and explains many of the terms. This one is also good.
TE connectivity has another good app note.
Your terms might be the same thing because it will cut off below hold, so perhaps there's confusion there. Usually the terms you're looking for is the difference between the trip current and the hold current, which is a hysteresis effect. The fuse will trigger at the trip current (lets say 0.5A) but once triggered, will stay triggered until the current drops to reache the hold or cut off current limit. This has obvious implications for the design to where you need to select those parameters to match what you expect will be short circuit or over current conditions. In PTCs the hysteresis has to do with the thermal design. Once hot (which is the way they work), it requires the temperature to be much less than the temperature caused by the trip current in order to reset itself.