When a particle is emitted from a nucleus, is it detected at only one point in space, or everywhere ?

Answer 1

It can be detected anywhere along the direction it was emitted in. It's not emitted "in all directions."

Particles are generally emitted with energies and momenta in a certain range, or with a certain probability distribution. There will always be some (linear) momentum assocaited with having that energy.

Because of conservation of momentum, the nucleus will also recoil. For each particle emitted, it can only recoil on ONE direction --- it can't recoil in ALL directions!

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