All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is not permissible to buy a house with a Riba-based loan. Naming the Riba-based interest as cost of borrowing or anything else does not change its ruling. Therefore, if the father was not in a dire need to engage in that transaction, then he should seek forgiveness from Allah for what he had done and be determined not to return to it again.
The prohibition of the Riba-based loan is related to the person who had taken this loan, and not to what this loan was spent on, like a house or anything else. Therefore, there is nothing wrong if the son, or anyone else, uses that house, because the prohibition is related to the person who borrowed the money, and not to the thing on which the loan was used. The house is the property of the father.
As regards the appliances in the house, water, electricity and gas, then this has nothing to do with the amount of the loan that was used for buying the house, as you mentioned.
In any case, there is nothing wrong in making use of the house and its equipment and appliances, and you do not have to limit yourself to the necessity or need. Rather, you may make use of them in a lawful manner without any harm at all.
For more benefit, please refer to Fataawa 124015 and 102369.
Allah knows best.