2004 Mitsubishi Lancer Heavy engine shaking!

"Doesn't start" can mean a lot of different things, but since a push-start worked, I'm going to assume the starter wouldn't crank the engine. Correct me if I'm wrong. That, along with the other symptoms points to a failed charging system and the battery has run down while driving. You should have seen a warning light on the dash or the "Volts" gauge would have been dropping. The headlights would have been getting dim too.

If all of this sounds right, start by measuring the battery voltage with the engine off and with it running. Off, it must be 12.6 volts. If it's around 12.2 volts, it's okay but discharged. If it's near 11 volts, it has a shorted cell and must be replaced. With the engine running you must find between 13.75 and 14.75 volts. If it stays the same as it was with the engine off, the generator, voltage regulator, and associated wiring are suspect.

Also, follow the smaller battery positive wire to the under-hood fuse box and be sure that connection is clean and tight. Those commonly work loose on all car brands.

OHHH! So NOW we get the rest of the story. You have another problem that has a real easy fix. When the battery is disconnected, the Engine Computer loses its memory. It has to relearn "minimum throttle" before it will know when it has to be in control of idle speed. Until then, it may not start unless you hold the accelerator pedal down 1/4", it won't give you the nice "idle flare-up" to 1500 rpm at start-up, and it will tend to stall at stop signs.

To meet the conditions to trigger the relearn, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.