Yet another Q about an old Windstar?

1995 Ford Windstar but the previous owner replaced the engine, (from a Taraus or something?)
Don't know if it was rebuilt or salvaged but I've owned it since 2006 and it is a champion. 3.8 liter V-6 put in sideways (Doh!) because front wheel drive.

At idle I'm getting a low oil pressure idiot light. I turned up the idle a little and it's been months since I;ve seen that light come on while waiting at stoplights but now it is happening again, on occasion. It's intermittent, only happens once in a while. Someone mentioned it may be a lose timing chain issue but that makes no sense to me, how that could be related.

Also, I have a secondary issue. Had the auto transmission completely rebuilt 3 or 4 years ago by a very reputable shop but now every couple weeks it has trouble finding and getting into 2nd gear. After a minute it is fixed and doesn't happen for another couple weeks.

All fluid levels are correct.
So what's going on? Is my oil pump failing? Is the trans getting confusing signals from the engine computer? I drive this car very gently and it seems that my feather touch on the throttle is when the tranny acts weird, but if I drive it hard it works fine.

Your opinion as to cause(s) and should I just take it to the Ford dealer and pay (use to be) $75 for a diagnosis? I can't get to the oil filter because it is up against the firewall.

Added (1). EDIT: 3d question: the darned warning sound comes on now that the weather is warming up, apparently it thinks a door is open. I have tried the Wd-40 on the sliding door contacts and that worked well all winter but now it's doing it again. How can I make that noise stop?

I know nothing of Ford AO trannies, I'm an old C3-C6 kinda guy but I'll try to explain the oil light timing chain problem.
The timing chain wraps around the end of the crankshaft and camshaft sprockets and they push a fiber shoe against it with an hydraulic cylinder to keep it tight. The hydraulic cylinder gets it's oil pressure from the engine oil pump. As the fiber shoe wears, the cylinder overextends and leaks oil. This, warmer temps and a tired motor and oil pump are "what I think" are causing your oil light at idle.
I think a Ford check makes sense, they usually give a repair estimate and you can decide if you can afford to keep it.
Junkyard motors of 3.8 ilk are usually in the 50-75 thousand mile range, so add whatever mileage you've driven to see how close you are to it's 150-200 mile life span…

Drop the transmission fluid pan (or have your mechanic do it). If there isn't an excessive amount of clutch material in the bottom of the pan, I'd guess your transmission problem is being caused by a bad shaft speed sensor and not by something mechanical. Those sensors tend to fail rather reliably on those transmissions… When the input shaft speed sensor starts failing (for example), it often causes hesitation during upshifts from 1st to 2nd.

As for the door problem, it could be any of the door sensors causing it. If you've addressed the contacts on the sliding door, then I'd guess the driver's door switch is in need of adjustment or replacement. It gets used a lot more than the passenger door so it usually fails first, but test them all until you find the culprit.

And when oil pressure sensors begin to fail, they can also cause the oil light to trip intermittently, especially at idle when oil pressure is lowest. You'd have to have the oil pressure tested at idle to know for sure if it's a faulty sensor or actual low pressure causing the light. Not saying that's your problem, but at least it's something to hope for.