MOSFET Switching without gate voltage / LED Strip

Point 1 and point 2 are quite accurate I would say. For point 3, I always search for the following graph when I'm looking for the likely conduction capabilities of FETs: -

It tells me that if I put 3.5V on the gate and I want to pass 10A drain current there is likely to be 0.5V dropped across the device leading to a power dissipation of 5W. This is an on-resistance of 50 m\$\Omega\$. However, if i used a gate drive of 10V, I can expect the voltage dropped by the device to be about 0.25V i.e. a power dissipation of 2.5W or an on-resistance of 25 m\$\Omega\$.

Regarding the state of "fully-on" you can see that there is no magic state that is reached at some arbitrary gate voltage but I'd say that at about 4.5V gate drive, the benefits of driving with a higher gate voltage are diminishing.

Figure 3 in the data sheet (which shows the graph which you deduced the on resistance variation of 10 m\$\Omega\$) is a derivation of the graph (figure 1) in my answer. Obviously, Figure 1 carries more information because it covers a wider range of gate voltages.

A single transistor will not provide an interface with your FET AND do it without inversion. You can do the job with two transistors, though.

^{simulate this circuit – Schematic created using CircuitLab}

This will provide a decent drive for your FET with about a 1 usec added to on-time. That is, if you put in a 10 usec pulse, the FET will be on for about 11 usec.

Also note that R5 needs to be a 1-watt resistor.

EDIT -

It turns out I was using the wrong FET model, and my value for gate capacitance was too low. For a "real" IRFZ44, the added pulse width is about 3 usec, not 1. The problem is that, when you go to turn off the FET, The charge stored in the gate can only discharge through R5, and the time constant is in the 3 usec range. You can, of course, decrease R5, but then the power it dissipates when the load is driven on goes up. You can get a 2 usec delay rather than 3 by decreasing R5 to 100 ohms, but then you need a 2-watt resistor. If you need very narrow PWM cycles (low effective motor current) you're probably better off with a more sophisticated driver. I personally tend to go with the Maxim MAX4426/4427 (less than $4 each, with 2 channels per IC), but that's just habit.