Have vehicle scanned at either auto parts store or reliable repair shop
A spark plug that is causing a misfire is one of many reasons for the check engine light to come on.
Transmission jumping hard is the malfunction causing check engine light to come on,most likely its the tcc solonoid causing this or a possibility of causing this.
If it is causing the cylinder to misfire, yes.
Take it to Auto Zone they do a free check on this problem and they tell you what is causing your light to come on.
That would be a good problem to repair even if it's NOT causing the check engine light to come on... but I would suspect yes, depending on the manufacturer.
it is probably throwing off a sensor, like the mass airflow or map sensor and causing the check engine light to come on
No, the check engine light does not come on because your gas is half way full. The check engine light will come on if there is a problem with the engine.
If an check engine light does not come on it is a good thing. When your check engine light comes on, it means you need service done on your vehicle.
No the check engine light will not come on
You do not need to fix the check engine light you need to fix the problem that is causing the check engine light to come on. Most auto parts will scan your ECU for free and tell you the code that has been set. This code will tell you what parts are being effected.
If you don't fix the problem that is causing the check engine light to come on you can unplug the battery a thousand times and the light will continue to come back on. Have the codes pulled and find out why it's turning it on in the first place.
A scan tool is required to read the trouble codes.