No, Hindu scripture does not discuss whether buying or renting a house is superior. But it does recognize both of those as legitimate activities. Here's what this section of the Brihaspati Smriti about acquiring immovable property:
Immovable property may be acquired in seven different ways, viz. by learning, by purchase, by mortgaging, by valour, with a wife (as her dowry), by inheritance (from an ancestor), and by succession to the property of a kinsman who has no issue.
And here is what this section of the Narada Smriti says about renting:
If a man has built a house on the ground of a stranger and lives in it, paying rent for it, he may take with him, when he leaves the house, the thatch, the timber, the bricks, and other (building materials). * But if he has been residing on the ground of a stranger, without paying rent and against that man's wish, he shall by no means take with him, on leaving it, the thatch and the timber.
Which sounds strange to me, but I suppose houses were more primitive then.
But concerning "Does Hindu scripture discuss whether the house one buys should be a new one or an old construction?", here is what the sage Bodhya tells Yayati in this section of the Shanti Parva of the Mahabharata:
I conduct myself according to the instructions of others but never instruct others myself. I shall, however, mention the indications of those instructions (according to which my conduct is framed). Thou mayst catch their spirit by reflection. My six preceptors are Pingala, the osprey, the snake, the bee in the forest, the maker of shafts (in the story), and the maiden (in the story)! ... Hope is very powerful (in agitating the heart), O King! Freedom from hope is high felicity. Reducing hope to an absence of expectation, Pingala sleeps in peace. Beholding an osprey with meat in his beaks, others, that have not found any meat, assail and destroy him. A certain osprey, by altogether abstaining from meat obtained felicity. To build a house for one's own self is productive of sorrow and not of happiness. The snake, taking up his residence in another creature's abode, lives in felicity. The ascetics live happily, betaking themselves to mendicancy, without being injured by any creature, like bees in the forest. A certain maker of shafts, while employed at his work, was so deeply attentive to it that he did not notice the king who passed by his side. When many are together, dispute ensues. Even when two reside together, they are sure to converse. I, however, wander alone like the anklet made of sea-shells in the wrist of the maiden in the story.'"
On a side note, this list of Gurus overlaps with the the list of Dattatreya's gurus whom I discuss here. That list was narrated by Dattatreya to the king Yadu, whereas this list was narrated by Bodhya to Yadu's father Yayati.