How feasible is is it to build a house structure from different sources of free timber?
Believe it or not, wood framing is a minor cost for the construction of a home. 

Yes, you “could” gather enough to frame a home and save money, but at what cost: 1) TIME: There is a huge amount of time to remove it from a site, remove nails, bolts, mastic, etc. so you can use it, 2) QUALITY: You’ll need to sort the material so you have adequate amount of good, straight, clean, sound material for structural use, (i.e.: headers, floor framing so no squeaky floors, etc.), 3) UNIFORMITY: Make sure the lumber is uniform in size so walls, roofs, etc. are straight. 

Meanwhile, while you’re saving a nickel on the lumber, I presume you’re paying rent monthly that could go towards mortgage payments if the house was finished sooner.