If a person pays monthly Hissa Amad and from that same income builds property, must Hissa Jaidad also be paid? Isn't that paying twice?

Yes, it should be paid. Hazrat Musleh-e-Maud (ra) answered: "The Wasiyyat is, in essence, to be paid after death from whatever one leaves behind. When a person saves money from his income and builds property with it — if, instead of the property, he had invested it in some other work, then whatever income came from it, he would pay Hissa Wasiyyat from that. So when he spends money on property, why should he not pay the portion from it, since it is property that has been created?"