You can ask far more than one question. =)
I think you have the TV show simplification of what "paraplegic" means, and probably don't understand spasticity too well either. Neither of those things are surprising, and even on this relatively educated website people don't use spasticity as its proper technical term very consistently.
Let's start with your ideas about spinal cord injury. People who become paraplegics due to traumatic, sudden spinal cord injury (accidents, that kind of thing), don't necessarily lose feeling the way you think. Different parts of the spinal cord carry different messages, and it is possible to injure the motor (movement) part and leave the feeling relatively intact. The worse the injury, the less likely that will happen, but there is a range of sensory loss from total to none at all. Paraplegia itself as a term refers only to movement, not to sensation.
That whole "paralyzed from the -blah- down" thing is also an oversimplification. Different muscles in the same body part are fed signal by different levels of the spine, so it is quite possible to have control over one set of them without their antagonists. C-6 quadriplegics are the easiest example, as they have biceps but no triceps. Those normally work together to make arm movements, but you can learn to use just the biceps and set your equipment up to use the motions they can make with strength.
For someone to be called a paraplegic, they simply to have some paralysis in the lower half of their body. If they have weakness but no paralysis, that would be called a "paraparetic" instead. HSP stands equally for Hereditary Spastic Paraplegia/Paraparesis. Most normal people don't know what a paraparetic is, though, so you'll forgive those who don't bother to make the distinction I hope.
The core effect of HSP is the slow death of the upper motor neurons in a particular track of your spine. That is what all forms of HSP share, it is the same track hit by PLS and ALS. That is the "UMND" part of it. This causes hyperreflexia, spasticity, and yes, eventually often total paralysis of a muscle group. It is a death by a thousand cuts, but if signal can't get through those UMNs from your brain at all due to the extent of the damage, you will lose all voluntary control of the muscle group. My first paralyzed muscle group was the one that you use to push the ball of your foot downward. Unlike someone who becomes paralyzed due to lower motor neuron involvement, not much atrophy happens. You can't get a signal to that muscle group on purpose, but your damaged UMN's are spewing static at it. That is enough for the body not to automatically use it for spare parts. In the traumatic spinal cord world, this is considered very desirable because it keeps you from getting "wheelchair legs" and also makes blood clots and some immobility damage less likely.
Some forms of HSP also damage the nerves of one of the spine tracks that carry sensory information. Normally they do this slower than the motor part of the disease, so loss of sensation doesn't match the speed of mobility loss. What kind of sensory you lose can also vary - pressure, temperature, tactile.
Spasticity is a big enough and complicated enough topic I'll give that its own post later. I hope this helps in the meantime.
