This is not a simple question !

I will deal with the easier parts first.

1. Your definition of ferroelectricity is incomplete. Yes, there should be a spontaneous polarization, but it must be reversible under a sufficient;y large electric field.

2. The difference between ferroelectric polarisation and dielectric or paraelectric polarisation is that in the latter two cases you need to apply an electric field to observe the polarisation. The main differences between paraelectric and dielectric polarisation is context - a paralectric is the state just above the Curie temperature of a ferroelectric material and as such will normally exhibit a temperature dependence of permittivity that obeys the Curie Weiss law, which is evidence of a soft phonon mode. GIven the temperature constraints of DFT, this is a difficult issue to address.

Back to your first question; with DFT you can determine whether your material has a dipole moment by summing the Born effective charges in the unit cell. However you need to be careful about whether you are looking a single unit cell or a large supercell to determine whether the dipole moment is coherent across all unit cells - that will then confirm whether you have a spontaneous polarisation and not an antipolar or randomly polar material. BUT proving that the spontaneous polarisation is reversible is the problem.

Essentially the definition of a ferroelectric is a phenomenological one - it relies on the observation of a field reversible polarisation, which does cause some issues. In experiments, you may have a material that from its crystal structure looks as though it may have a spontaneous polarisation, but for which the coercive field is beyond the breakdown field of the material. You cannot prove this is a ferroelctric.

However your problem is slightly different and comes down to how well equipped is your implementation of DFT to calculate structures under applied electric fields, i.e. so that you can prove reversibility of the dipole moment under a sufficient;y high field.