Air carrier First Officer reported confusion and miscommunication with the ramp crew about which engine they should start first.

Narrative

APU bleed air was deferred and inoperative; requiring an air cart to start the engines. We confirmed with the rampers that we needed an air cart to start the engines at the gate. Once ready to start; communication with the tug driver was established via intercom headsets. We were given the clearance to start at the gate by Ramp Control and the tug driver signaled clear to start. The other pilot told me to start engine number 1. After engine start was complete; there was lots of confusion on the Ramp and the tug driver was not answering our questions or responding to hand signals. After no response from the tug driver and talking with Operations on the radio trying to relay what was going on; we got no helpful information from anyone outside the aircraft and with the cabin temperatures rising to 90 degrees; we elected to shut down the running engine. At that point the Captain stepped outside to talk with the ground crew who had all gone into their hut. After a quick discussion; we were once again communicating with the tug driver via the headset and started engine [number] 2 with the air cart; and were on our way with no further confusion. Lack of communication on which engine to start first. [We should] confirm with the ground crew early on when asking for the air cart what engine we expect and they expect to start first.