A lot of things can go wrong. Lug nuts (that hold the tire to the wheel) may not have been tightened properly.
The studs (threaded bolts that the nuts thread onto) may have sheared off. RARELY happens!
The wheel bearing could have came apart from not being caught or repaired when failing.
The bolts that hol the wheel bearing to the car may have broken.
or many other possibilities. Has the car been in a wreck??
Improper installation. Lug nuts were loose.
To the right
most likely the car's tires are out of alignment, or you have a slack tire.
Yes, either can cause a shake while driving.
Heat is built up while driving and that will increase pressure in the tires.
Immediately pull off the road and stop. Set the parking brake on your car. Use the jack to jack up the car and change the flat tire to the spare. If you are unable to accomplish this call for help.
the friction between the tire & the surface your driving on causes heat and which heats the tire + the air...
Tire out of balance, bent wheel, loose wheel bearing, defective tire, loose wheel lug nuts, & worn steering or suspension parts.
Most likely it has unbalanced or bad tire(s). Bent wheel rims also can be the cause.
Of course you can, if you are a total nutcase...
The inside volume of the tire has not changed, so the pressure will not change.
car starts but shut off while driving