'99 Corolla, (Inconsistent) Starting Problems

#1
We have a '99 Toyota Corolla, and love that car. It has about 140,000 miles on it, and runs perfectly (we drove it through the US a while ago). However, in the last weeks it fails to start once in a while, generally after it has been used for a while, and sat for maybe 20-30 min. in some parking lot (when running errands). It is rainy those days it does not start, it seems. The starter turns over, and it even runs briefly, but then dies. After several tries it does not even turn over anymore. When the car is being jump-started, it works (surprisingly).

What we did/ looked into up to this point:
- battery in good shape: tested it when car running, starting, and even when car does not work: battery also good;
- no codes come up (but only done when car works)
- cleaned/ checked for water engine: none;
- changed spark plugs a while ago: all new now;
- oil at perfect level;
- let our mechanic check all electrical connections: cleaned it, and did not notice anything unusual;
- using injector cleaner on a regular level (given filter/ injector within these models), and also just put in higher octane gas;

It sounds like an electrical problem, since the car still runs perfectly otherwise. It does start every morning (it is sitting in the garage over night), and runs perfectly (sometimes a little weaker, it seems, and there is a fear that it might turn off when sitting at a stop light, at times: but that might be our sensitivity at this point - it has not turned off yet in such circumstances). Such issues are impossible to find though - or does anyone have an idea?

My other guess: had mechanic not put oil cap on after oil change about 1.5 years ago, and still have oil on insulator hood. Could that dripping into engine area cause it? Moved to more humid climate (Arizona --> East Coast).

We do not want to let this car go but we need it to be reliable!
Any help/ suggestions? Electrical? What to check? What to change first?

THANKS!
 
#2
When it doesn't start do this next time. Turn key on after it doesn't start, do you hear a faint humming coming from the back for about 2-3 seconds? If not then fuel pump is going bad, replace distributer components, new cap rotor, igniter ect ect, I'm not sure what parts go bad on that engine but seems it's when it's hot from running that something isn't making contact.
 
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