Check engine lights can come on for a number of reasons. If you didn't get the cable right or got a sparks when connecting or disconnecting that could most definitely cause problems with the check engine light or even blow the alternator fuse.
a weak battery will not cause an engine vibration.
A dead battery would not cause an engine to seize
Battery acid does not build up, there is a fixed amount. Provided the battery is kept topped up with pure (distilled) water to replace any lost gases during charging, it should last a few years. There should be no battery acid in the engine. The engine needs battery power only for starting.
No, a defective battery will prevent the engine from starting but once it is running the battery will not cause hesitation. the the starter relay is disengaged , and the alternator takes over battery charging via voltage regulator . voltage regulator prevents power surges and prevents voltage fluctuation.
The battery light can be on for several reasons. A bad alternator is the top cause in my experience. Have the charging system tested and see what is going on.
The most common cause of automobile starting problems; is the battery needs to be charged. When the battery will not take a charge. It needs to be replaced.
Dead battery
Low battery, low fuel pressure, a code stored in the engine computer that will point to a possible cause. Plugged air filter, internal engine problem,
After repairing the cause of the code, you can clear it with a scan tool.
A Low battery could cause this - try getting a jump from someone,when it is cold the battery will, if they are low sometimes, not have the cranking power needed to turn over a cold engine. To test if your battery is low turn on your lights if the lights are dim then it is your battery.
The alternator produces power to charge the battery. The battery powers the starter to turn the engine over.
try checking the battery cables