""Does it mean that getting a wife must be strictly timed to coincide with getting a house?"
Essentially yes. A man must have his own dwelling so that he may start a family. The "timing", structure, and living arrangements vary from culture to culture.
In this country, some parents can't get their 30 year old sons off the couch and away from "Call of Duty" long enough to get dressed and get a job. In the third world, children are often put to work as early as possible. My family traditions are full of such tales."
"while marriage assures that the child bride has a house"
There are no assurances in life, especially in third world cultures."

I mean, assurance because of requirement that there should not be two Mrses in one house. If it applies, then a 30 year old son may have been working hard for 20 years... but is it a requirement of getting married that he should get a house shared only with his wife and NOT with either his father or his mother?

""Dowry" is an old word, while "bridewealth" (also known as "bride price") seems to be a newer term."

Because the English had a dowry system and therefore had a word for it. They did not need a word for bridewealth till they met people who had it.

But that´s a big difference. In case of bridewealth system, a family in economic crisis would have the incentive to sell daughters off younger than planned and at a lower price, for immediate needs. In case of dowry systems, a family needs to pay to marry daughters off and economic crisis means the marriages are delayed rather than brought ahead - the families get stuck with old maids.