Not true, though the exact boundary is not well defined. See US v. Causby

https://en.wikipedia.org/wiki/United_States_v._Causby

Depends where you are, but generally you are good to around 400ft above your own property in most cases.

FAA jurisdiction over the airspace is not the same as a rule prohibiting every form of powered lift-off within it. There's a lot of grey area in FAA regs - for instance you can land a plane on your lawn in class E (if you could manage it).