Q: Battery/alternator/carburetor - anything in common? 2004 Nissan

Last weekend, I left the light on in my car and Found that the battery had died when I returned from a weekend traveling. I jumped my car and went to work. I had to jump it again to get it running as the battery had died again. I decided to take it to the auto shop to check if the battery was bad. They claimed both the battery and alternator was bad. Although my coworkers said it was unlikely that both woukd go out, I decided to give the mechanics the benefit of the doubt and replaced both the alternator and the battery. Before they would replace it, they made me come into the shop to show me that my car 'rough idles'.(Just vibrates a little when starting the car; very minorly). When they returned the car, it now won't start unless I push on the gas petals when starting it. Otherwise it stalls & shacks & vibrates violently when idling which it was never done. Another mechanic said the caburator was the problem. Could the 1st mechanic have caused the problem? They have a bad reputation

My car has 86000 miles.
My car has an automatic transmission.