Shocks

If you replaced them with shocks that are like the originals then you should not expect any lift. What would jacking it up do? Do you mean installing a lift? The leaf springs are what really hold the weight.

Absolutely. If you want a truck to ride like a car, you're going to get the same weight-carrying-capacity as a car, (meaning none). Springs have to be really stiff to hold up the weight of a load. Front coil springs provide a better ride than the leaf springs often used decades ago, but the front doesn't have to work as hard as the rear.

A compromise is stepped rear leaf springs. Those use only half of the spring so the ride is smoother, but as weight is added and the rear settles, it hits the second half of the springs which increases the load-carrying capacity. Some trucks, particularly Dodge dually diesels, come with those springs. An F-150 is a relatively very light truck so it likely has standard rear springs. You might investigate at the dealer if lighter springs, or stepped springs are available. With lighter springs, there are also aftermarket air springs that can be added. Some of those kits include an on-board compressor, and control and pressure gauge inside the cab.

It's "quad suspension" not "quad cab suspension"

It means you would have 4 shocks in the front instead of two if you have the quad package.

You could have bent a wheel or ruptured a steel belt in the tire. Run the car with the wheels elevated and see how true they spin.