Pruning can be healthy for your cactus, if done right. Removing pads that are growing too close to each other helps to prevent overcrowding and gives the rest of the cactus more energy to grow. This type of pruning is recommended as most cacti (with pads) grow somewhat 'thicker' in gardens and optimal growing conditions make overcrowding of pads a common issue.
Pruning to style your cactus — while not necessary — can be fun and give you some good looking results. However, you have to keep in mind that the shape and size of a cactus has some natural per specie limitations. You could get a cactus to grow taller by removing lower pads but depending on the species it might not look — or be — healthy as a tall plant. As a rule of thumb; if it does not have a 'trunk' (only pads growing on top of each other) then it will be fragile when forced to grow taller then it naturally does.
If you are intent on growing it as a tall cactus, try keeping the weight at the bottom of the plant. Keep the bottom pads bigger than the weight they have to carry, this way it doesn't become top heavy. Example:
No, there should not be any negative effects from removing pads as long as you use a clean pruner.
Yes, the pads can be propagated (used to start new cactus plants), and this can be useful as you then have some cacti to 'experiment' on.