As far as the purge solenoid valve, it allows gas fumes to get burned through the intake manifold. Its part of the emissions control system. This system's goal is to prevent gas fumes from entering the atmosphere. I hope that answers that question for you. And yes, the hose should be replaced. Has it been missing for a long time?
As far as the hard start, it could be the result of several things. The first thing I need you to check is fuel pump pressure. If the pressure is too low, it can cause a problem. However, if it is too high due to a bad regulator, that can cause issues with the injectors.
If that checks okay, then the idle air control valve (IAC) needs to be checked for operation. The IAC is responsible for maintaining the engine idle speed. On a fuel injected engine like yours, air is regulated by the IAC that enters the intake and engine. If the valve isn't opening properly, that may be the reason you have to press on the gas pedal.
If that isn't the issue, we will need to check injectors for leaking, compression, coolant temp sensor. All of which can cause this problem.
NOTE: Let me know if you see any black or white smoke when it finally starts for you. That can be helpful.
Saturday, June 21st, 2014 AT 6:56 PM