The oil can be a problem if it is low ,when starting cold their is alot of friction causeing alot of load which combined can cause the engine to stall
the alternator can also cause a large load causing it to stall but unlikely
Your problem goes away with you pressing the gas pedal ,so it meand you need more air to keep the engine running
sounds liek a sticking iacv
get a motorvac and make sure you iacv is working