Assuming I don't care about moire and would like to have my images as sharp as possible, would it make sense to remove the antialiasing filter from my DSLR? Is it something that can be done at home or do you need specialized equipment?
No, it would not make sense to do this.
The first reason is because the optical low pass filter (OLPF) and sensor are often glued together to help with flare issues, making this operation impossible for many cameras.
The second reason is because even on a camera that it could be removed on, you will change the filter stack thickness doing this, reducing image quality for wider aperture and wider angle lenses.
The third reason is because even if you manage to remove the OLPF without scratching the sensor (a big if, bare silicon is easily damaged), putting your sensor back into place well enough aligned to not cause a blurry edge requires precision to just a handful of micrometers. This is not a precision readily achieved without specialized tools.
It can if you replace it with clear optical glass having the same refractive index and thickness.
It's also pretty much an absolute certainty if you had the knowledge, skills, and very precise equipment needed to properly do such a job you wouldn't need to ask the question here.
It just would not work. If you remember the D800 and D800E pair, the latter had a layer to replace the anti-alias filter since otherwise the position of the sensor would be off because it has been placed with the assumption that there is a filter in front of it and its refractive index is not equal to air.
Yes it would make sense to do this (if you can do it yourself correctly) and that's the only hardware difference (I do not recall any exceptions) between usual cameras and specific models without AA filter (i.e. AA filter gets replaced with clear glass in these special models).
The effect depends on the model, there is a fairly large number of cameras with weak AA filter (many of Nikons for example). Additionally, conversion is offered as a service, by this company for instance.