What happens with the electrons is generally ignored because it has no bearing on what happens to the atomic nuclei. By the time any interactions with the electrons might occur, the nucleus has already separated. Two electrons might come along with the alpha particle to form a helium atom, or the end state may be a pair of ions, or some number of electrons may be knocked loose and go elsewhere. Regardless, they have little effect on the alpha particle, and typically the focus of attention and study relating to alpha radiation is on the alpha particle or other atomic nuclei.
As for why it's specifically an alpha particle that separates from the nucleus:
An atomic nucleus is stable because its composition and arrangement form a local minimum energy state. 969
If the nucleus were a macro scale object, classical mechanics would require that in order for it to change, such as by emitting some number of separated nucleons, it must first gain enough energy to climb out of that local minimum and cross a nearby local peak energy state on its way to a new and different local minimum. However, an atomic nucleus is actually small enough that quantum mechanics have strongly relevant effects, and quantum tunneling allows the nucleons to randomly skip past such a peak on occasion.
Even with quantum tunneling, however, energy is still conserved, so the end state has the same overall energy as the initial state. The energy of the remaining large nucleus, plus the energy of the emitted particle(s), thus must be no greater than the initial energy of the pre-emission nucleus.
An alpha particle has an exceptionally low amount of energy for the number of nucleons in it, and emitting one also has minimal effect on the ratio of protons to neutrons in the remaining nucleus. These factors combined make the total energy of an alpha emission end state low enough for it to be reachable from the nucleus's ground state without violating conservation of energy.
Emitting a proton, neutron, deuterium, tritium, or helium-3 nucleus may be theoretically possible, but would require a higher initial energy state. Such high nucleus energy states generally only occur when another particle collides with the nucleus, and electromagnetic forces tend to prevent that in normal Earth environments. 362