That's a very interesting response. I find it highly unlikely she made it to age 9 without ever hurting anyone or stealing anything. Without seriously hurting someone and without stealing anything of significant value, sure, but children hurt each other and filch cookies all the time. On the other hand, that obviously wasn't what Alexis meant by the question. The fact that she's able to answer 'no' implies the Zone of Truth works on her understanding of the question rather than compelling literal truth. That's a powerful advantage - it probably means ZoT is immune to the 'Truthful fact presented deceptively' trick - but does mean you need to be careful about phrasing your questions to avoid misinterpretation or false assumptions.
'Homunculus' is the singular; the plural is 'homunculi'. Also, it's not a proper noun, so it shouldn't be capitalized.
This is all based on an incorrect assumption: the potable hole trick doesn't offload the weight of the structure; it offloads the force of gravity. The structure (with the possible exception of the part that is physically within the hole) is still subject to gravity, which generates a downwards force; this force is transmitted through the support beam, into the anchor inside the hole, and ultimately into the walls of the hole itself. So, for example, buoyancy isn't a problem; buoyancy will produce a small upwards force, but, since the structure is denser than air, this force will be much less than that of gravity; the net force will still be strongly downward, and will still be transmitted into the hole. (Conversely, if the structure were lighter than air, e.g. a helium balloon, the net force would be upwards, and would try to pull the anchor out of the hole instead.)
Actually, other than overcoming air resistance, the horizontal force will be zero. The whole point of the exercise is that forces do not transmit between the walls of the hole and the external anchor; that's why the inside of the hole can support the force of gravity from the whole structure while the flying carpet only has to support the weight of the metal-plate-with-a-portable-hole-on-it. As far as the structure is concerned, it isn't accelerating at all; it's the rest of the world which is moving. So if the carpet is flying 50km/h, the support beam and anchor need to handle the force from a 50km/h wind, but that's it.
... which, incidentally, means Alexis was wrong about needing to keep the weight down in order to let the carpet accelerate quickly. Consider the carpet being ridden normally: when the carpet tries to move, it generates a force which is applied to itself and, assuming they are holding on tight, the passengers. Acceleration = Force / Mass, so if you add more mass, it will accelerate slower. But, in this case, the structure isn't supported by the carpet; it's supported by the inside of the hole, and forces cannot transfer from the carpet to the hole. So when the carpet starts moving, one of two things must happen: either the structure does not move at all - in which case the whole idea is dead in the water - or the structure moves along with it without any force being applied to it - in which case the mass of the structure is irrelevant. Of course, the the carpet isn't really moving the structure at all: at all times the structure remains stationary relative to the inside of the hole, and the carpet is simply moinvg the point in the outside world where the hole happens to be.
On the other hand, holy water IRL isn't a substance with verifiable magic properties.
Abuse of authority is an evil act even if it isn't a criminal one; you might argue her response was disproportionate, but it wasn't entirely unjustified.
You are correct that none of his actions are ones which would be worthy of execution in the legal justice system, but they would certainly deserve incarceration. Unfortunately, his status as a zombie means the legal justice system is unable to effectively deal with him, which places him instead in the jurisdiction of the Slayer. And, as Giles points out in the chapter, they don't have the capacity to imprison him. The only practical options are to either kill him or let him walk free.
Now, the fact that Alexis was advocating for his death before Giles listed his crimes, when all she actually knew was that he was a sleazy douche, that is ethically problematic.
They are operating under military-style rules of engagement rather than police-style. They don't try to check on any individual basis whether a given demon is evil; instead they judge by species, and possibly visible symbols of allegiance or intention, because most demon species are, in D&D terms, Always Chaotic Evil. You don't need to investigate whether a given vampire has actually eaten anybody; the number of vamps that aren't lethal threats to any human they can catch can be counted on one hand. We're informed that there are some species of demons that can coexist peacefully with humanity; even if it never comes up on-screen, presumably a member of these species could walk right by a slayer without more than a glance. Scoobies may tend to assume that a demon of unknown species is hostile until proven otherwise, which is unfair, but that's not unreasonable given the odds.
Now, what would be interesting would be how they deal with demon species that are neither universal threats nor generally harmless. If there's a demon species where a substantial portion of the population live peacefully but a substantial portion kill people regularly, do Scoobies switch to an investigate-on-individual-basis system? Or do they just slay 'em all and let hell sort 'em out?
Why bother with an electric motor? If you have steam, just route it through a turbine or expansion cylinder and use that to drive the wheels. Before internal combustion dominated, there were plenty of steam-powered cars. Indeed, they can be mechanically simpler than internal combustion cars: because steam engines can produce high torque across a broad range of speeds and aren't subject to stalling, many had no clutch or gearbox; the engine was connected directly to the drive wheels.
Holy water doesn't poison vamps; it burns them like acid. Tranq darts only hold a tiny quantity of liquid; while injecting it will probably do more damage than spraying it on their skin, it's not going to be fatal or even debilitating unless you get very lucky.
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