Is it permissible for a father to register an apartment that his youngest son built with his own money in the father's name, after the second son refused this, citing discrimination, knowing that the father built apartments for his other sons?

The father must be just among his children in providing for them according to their needs. Anything beyond that, such as a car or an apartment, may be lent while its ownership remains with the father. If he gifts something to one of his children, he must be just by giving the others the same, with the female receiving half the share of the male, unless the adult and mature children consent to the preference. The basis for the obligation of justice is the hadith of Nu'man ibn Bashir: "Fear Allah and be just among your children."

As for the apartment, your father's permission for you to build on it might be a preference for you over your siblings, unless they agree to it. Otherwise, your second brother must be compensated, and more importantly, the daughters must be compensated.

The children must gather to reconcile the matter, or seek arbitration from experts.

If the father can give everyone an equal gift, including the daughters, then this is what is required. If not possible, the value of what each of them received from apartments or gold should be considered, and all gifts should be redistributed according to their values, with the male receiving the share of two females, and whoever took more must return the excess.

There is no harm in the father gifting an apartment to his wife.

Care must be taken to preserve harmony, show dutifulness to parents, avoid injustice to daughters, and protect the father from such injustice.