2002 Pontiac Montana IDing overheating problem

PROBLEM - OVERHEATING:

1. Wondering if I have a "fan relay problem", or "fan problem" (like with just one fan?) Vs. Gasket problem?

2. What does it mean if a fan feeds back through the system? I saw this on a NAPA parts site for "electric cooling fan relay". Could this be the "problem"?

3. Could it have to do with the metal thing that attached to the water pump?

4. Could one fan be kicking in - while the other isn't kicking in -- and if so, is this a relay problem?

5. I looked up fans for Pontiac Montana. It says there are two types - one for driver's side and one for passenger's side. I only thought the fan was on the passenger side - so confused.

SYMPTOMS:
* Van with engine running, most of the time the heat won't kick in for about 30-45 minutes, then it'll be like the fans "cough" and then all of a sudden there is a nice blast of hot air, and the heat stays on.
When heat kicks in, van doesn't overheat.(Needle down to normal level).

* Van can be forced to "overheat by revving engine - Pontiac Dealership Service Dept. Did it.

* If parked and idling, it doesn't overheat. Drive and sit at a light with a warm engine, it will overheat.

* It doesn't like moderate to steep hills - up or down. Up - is worst. Will overheat.

* If system is bled, can be fine for a day or so, then problem reoccurs.

TESTS from my research:

Oil is clear, just starting to look like it will need to be changed (it's due in 500 miles.)

Coolant "doesn't splash out" of rad (coolant at neck level before engine start), on start up.

Coolant still is at full level in overflow tank.

A: I'm not sure if a hydro carbon test was done.

B. I don't know if these garages did a pressure test. Back in October when first picked up van out of second storage, Shell Station (here) did a pressure test and it tested "fine".

HISTORY:

I love my van; acquired from dealer in November 07. It did a 12,000+ mile road trip then developed overheating problem intermittantly, which has gotten increasingly worse. The problem FIRST surfaced AFTER being stored for three months outside - we had a lot of rain and cold in Dec - March 07-98.

The first signs of a problem after starting to drive the van again was the sytem light would also be on for about 3-4 minutes on start up, then off for remainder of any outting, until the engine was shut off and then on restart, the light again would do the same thing, etc. It was not a serious problem then. And the issue at the time was whether to store it - again- or sell it. Opted to store it / table the decision. So, it was stored from July - October 6, 08.

Picked up van. Engine light repeating the same as above. Van started having an overheating issue - which had only happened once in about May 08. Picked up van in WA and drove up fine to my city, then within two days the overheating problem surfaced. When I picked up the van, the coolant level was normal, then when I checked it the night it overheated, the overflow tank was dry. BUT then I filled it with Dex-Cool, and since then it hasn't changed levels (i.E. No leak).

After filling with Dex-Cool, went back down to WA (bigger cirty)*, no problems, drove a lot, no issues. Maybe drove gee, 160-200 miles? Drove back up - overheating issue resurfaced, this time on coming up a moderate hill near home. This time overflow tank is full. Starting to notice heat is SLOW to 'kick in'. Strong fan - but blowing cold air.

[* I can tell you privately where I am - I just don't want it published on the Net.]

Someone told me try putting on "defrost" "5" (highest blower setting), and highest heat setting. Did 'that'. Seemed to work initially, to keep it from overheating when it started to have the needle rise.

Garage #1:
Too into my very (!) Trusted mechanics garage, and it would never act up at the shop, so problem couldn't be isolated. Mechanic (John / Owner) would bleed out the line, thinking maybe air had gotten in somehow? Van would be "fine" for 1-2 days, then problem would repeat - while it would idle "fine" for over an hour at a garage when shop mechanic would test, for me when I'd pick up the van, drive about 5 miles, problem would resurface - idle at a light and it would start to overheat, or on an uphill. Second time in shop he put a new thermostat in it, and did tune up, checked hoses, wires, etc. Said waterpump looks okay. Belts are "fine".

Pontiac Dealership Service Dept:
Took to Pontiac dealership and the guy there put a scope on it. Says 3rd cylinder is misfiring, reading a problem with oil pump gauge (not diagnosed) and the fans couldn't be controlled with the scope - and said they should be able to do this. I notice some white smoke out of back at the time - on a cold morning, with engine cold. Pontiac said it was "nothing" and normal for cold engine and colder weather. Pontiac concluded "no diagnosis" - except - check fans, and oil pump (or pressure?) Gauge - and find out why 3rd cyl. Is misfiring -- then if not this, then gasket problem.

Pontiac was able to "force it to overheat" by revving engine.

I drove it home - about two miles - and it kept trying to overheat, especially idling at the traffic lights.

After it had been into #1 garage twice, (had been towed there twice), I took to #3 diagnostic at another trusted mechanic. It was towed there after overheating about one mile from home.

(Should mention, I volunteer for a nonprofit organisation - so Mechanic Shop #1 donates all their labour. #3 is helpful and trusted but no longer donating time. So need to be very conservative about how I spend our funds).

Garage #3:
#3 shop owner "Alex" after having one of his shop guys putting an hour into it, started thinking it is likely the head gasket. Said intake valves were done. Said water pump looked okay.

SHOP RATES/ LABOUR TIME QUOTED:
Rates: I am being quoted $79/ hour shop rate and 18 hours for labour, + parts. Estimate is about $2500 to do the job. How much should I be expecting in labour hours and if it is a blown gasket, what are normal rates for machining / grinding of flat heads?


QUESTION: Can a van in this condition be "traded in"? Our nonprofit isn't in a strong cash flow position to deal with trading in - so would like to fix, if possible, and it makes sense to do so.

Thank you for your help! Appreciate your answer.
Making donation - based on what our nonprofit can afford presently.