The engine symbol comes on and stays on. The car starts to vibrate when going over 40 miles per hour. The car is hard to crank sometimes.

360 km/hr is NOT 40 miles/hr.
1 mile is about 1.6 km
To go from km to miles, multiply by 0.62
To go from miles to km, multiply by 1.6

Kilometers per hour to Miles per hour

if the wheel alignment is faulty ( castor wobble ) at 70 mph there is no way that you can describe it as a vibration as it virtually rip the steering wheel out of your hands. If it is a torque converter problem then at around 3850-4025 rpm stationary there will be a vibration of the vehicle and engine The sort of vibration that you are experiencing at 70 mph will come from a drive line problem. If rear wheel drive then it could be faulty "U" joints, worn slip joint or failed rear engine mount
If front wheel drive it may be cv joints or bent drive shafts,wheel bearings or brake callipers not moving freely on the mounting bolts.

Do you have a check engine light blinking when this problem occurs? I assume you do because if it is missing and sputtering like you say the engine control computer would detect that and turn on the check engine light. Next step is to have the systems fault codes read through the diagnostic port under the dash in front of the driver, Autozone and others does this test for free.

IF you have rear wheel drive then check the u-joints for wear/loose. Check slip joint for wear . Check rear engine mount is in good condition. Check wheel balancing

How bad is the vibration? If it's a bad vibration it could be a faulty CV joint or word U joint.

Get the codes pulled, sounds like you are in limp mode and I am guessing an issue with the transmission.

You have it backwards

What you hear is engine noise from
higher rpm, because your trans is not
upshifting, as a result of your input on
the gas pedal

You need to have a trans or repair shop find out why
you have a delayed or no shift condition

When you get to 4th or 5th gear (overdrive) you also
get torque converter lock-up & things should get quieter

Yeah you may need a whole new dashboard because the whole panel circuit for may be messed up.

It could be worn starting parts, Not ballanced wheel, and broken or worn struts. Start with the wheel getting ballance and go from there if there was no problems before your got the new tires put on.

check engine codes. also may be a bad coil. you have 2. one on each side. it will run with 1 but vibrates very bad.