Does the person A own house D, in legal terms?
The legal owner of the house is entity C. But, person A is described for some legal purposes as an "equitable owner" or an "indirect owner" of house D.
For example, in international taxation, indirect ownership can often be considered for purposes of determining if a company is a controlled foreign corporation. Likewise, indirect ownership of entity C is relevant for purposes of federal diversity jurisdiction if entities B and C are LLCs rather than corporations.
Can the person A live in the house D without a rental contract from
the business entity C?
Person A does not have a legal right to live in house D without a rental contract or other permission from entity C. But if person A does so anyway, only entity C would have standing to sue person A in most cases for doing so, and person A could exert their indirect control of entity C to cause entity C to decline to bring suit to enforce its rights.
Person A could also use person A's indirect control of the entity C to cause entity C to ratify person A's use of house D after the fact on whatever terms and conditions person A desired (although this might not suffice if done after an alter ego veil piercing claim is asserted to defeat the alter ego claim).
However, if entity C does not enforce its rights and gets nothing in return, it has arguably made a fraudulent transfer of the fair market rental value of house D to person A during the time it was occupied by person A.
In an extreme case, person A's failure to have an agreement or express permission from entity C could even be a ground upon which a creditor could pierce the corporate veil and ignore the existing of the entities B and C argument that person A's disregard for corporate formalities has the effect of making entities B and C the alter egos of person A.