Vehicle stays on after key removal also intermittent no start issue?

Greetings esteemed automotive experts,

I write to you in hopes of acquiring professional insights into an ongoing issue with my vehicle listed above, a vehicle which has provided dependable service over the years. I have always ensured diligent maintenance; however, recent malfunctions have presented challenges that elude resolution.

Approximately one year ago, my car began exhibiting signs of battery drain. It required frequent jump-starts, prompting me to replace the battery. Upon further drain incidents, I returned to the battery retailer for inspection. No defects were found in the new battery; rather, the car radio was identified as the source of the power draw. The radio was consequently disconnected, and the battery drain ceased.

Fast-forward a year, with the radio still disconnected, my car now presents a starting issue. After being parked for two to three hours, the car struggles to start. The engine appears to labor, attempting to turn over but without initial success. The interior and exterior lights, however, remain constant and do not dim during this process. The car usually starts on the second or third attempt, and, in rare cases, a fourth attempt is needed.

Simultaneously, my car occasionally continues to run after being switched off—one out of approximately ten instances. The engine remains functional, even allowing for revving. However, upon pressing the brake pedal, the engine shuts off abruptly and the vehicle shakes considerably. Notably, these violent shakes also occur when the car successfully starts on the second attempt but are absent on third or fourth attempts.

Another potentially related observation is a clicking or clinking noise originating from the gear shift area, primarily while braking. It's unclear whether this is connected to the other issues, but it seems significant enough to mention.

Lastly, despite recent brake disk replacements, I've noticed that the car sometimes exhibits prolonged stopping times. Following a stop, the car will idle for about a minute before the engine begins to shake, as if under some form of strain or distress.

I am eagerly seeking your wisdom and guidance to diagnose these issues. Any suggestions on potential causes, possible fixes, and preventive measures would be greatly appreciated.

Thank you in advance for your time and expertise.