What is the problem with my Honda Element?

When turning it see if a red or green key light turns on. It could be something as simple as the microchip in your key is going bad. If your element is acting like it is about to turn and just won't but after 5 or more minutes of just cranking the heck out of the key and pressing the gas make it start then it easily is just a key and a $25 fix to get a new one.

check again the connection of fuel pump. check and get tested the main relay/fuel pump relay

Check you air intake and tubes. I would have to warm up my '05 Element till my rpm idles at 1000 or below. I sometimes have to lightly press and hold the gas pedal on colder days. I cleaned my AIC valve watching a youtube video and purchasing cleaner solution ($5) at an Auto Zone. It sorta helped, but I might need to get the part replaced altogether.