Wellnow.
1) True
2) True
3) True
4) Not necessarily true, as it requires effort and more than one try. So, not often used.
Yes, it is an “American Suzuki,” if there is such a thing. And the parts are very expensive, which is why I have gotten them at Auto Parts Warehouse.com
The car started better on the second try at Walmart (when I came home) with my foot to the floor.
Again, I could have had this diagnosed properly in person, had I had the money for it, this morning. But now at least, it would seem we have “ignition” ruled out and “fuel” ruled in.
Now to get it looked at...
OK.. since #4 is not a ‘viable workaround’ that removes that puzzle part and makes everything a lot more consistent.
Revisiting the ‘pedal to the metal’ flooding fix, if flooding is the issue, it will still take a bit of cranking to clear out the extra gas, but opening the throttle all the way should allow it to start a bit more reliably. If you find that doing that lets it start ‘easier’ than it was you can probably tell your ‘fixit’ guy that the issue seems to be flooding or similar. Also if that works it will allow you to use the vehicle until you can actually get it looked at. Of course, if that is the issue, your fuel mileage will suffer.
If you know your way around under the hood at all see if you can locate that fuel pressure regulator. It looks like is would be mounted on the end of the fuel distribution pipe that goes to the four injectors (again, assuming that you have port-type FI.) If it really is there, getting the vacuum hose off to check like the second comment says should be easy. That part is showing up as listed for between $70 and $200...the high price is probably dealer.
Someone left some cupcakes with chocolate frosting sitting in front of my hot-caffeinated-beverage making device..