When it starts, does it start well? ie, starter cranks strongly, or is it labored?
Could be a number of things:
- weak or bad battery, especially if the battery is approaching 5-7 years in age
- heavily corroded battery terminals, pretty common, if they look "fluffy" - they can cause all sorts of issues
- bad starter contacts, corrosion in this area is pretty common, ensure the terminals are clean and tight
- bad ignition switch, less common but can happen, more so if you hang a lot of weight from the ignition switch, can wear out the cylinder
- could be bad chassis grounds, can be intermittent like that - same as the starter contacts, make sure they are clean of corrosion and on tight
- remote possibility, bad ECM - there was a number of Corollas in this particular generation and model years that had a defective board in the ECM - would cause the car to be hard to start, some cases, cause the car to just die without warning. There was a recall that was released for those models, but some owners that were not on the original list also had similar issues - which were fixed on a case-by-case basis. Most of the models that were affected had the Delphi made ECM instead of the Denso made ECM. Your ECM is right below the glovebox, you might make out the model if you stuck your head down there with a flashlight.