wrote

... a home (built in 1955) in the Oakland Hills (northern california)

My Canyon (2 miles from Oakland) CA friends don't believe in insulation,
nor anything else in the building codes :-)

My fianc=E9 and I recently found out that our monstrous 50 year old 255K
BTU furnace (70% efficiency in its day) has cracks...

.... 255K Btu PER HOUR seems like a lot. NREL says 1050 Btu/ft^2 of sun
falls on a south wall on an average 48.7 F January day in San Francisco
with a 55.6 daily max. ASHRAE's 99% winter design temp is 35 F, ie it's
warmer 99% of the time. Solar house heating is _extremely easy_ in SF.

... How is one to know who to trust and believe?

Trust and believe me. Stretch and Turtle are OK, but orthodox. SQlit
is dogmatic, mransley sometimes acts like an arrogant idiot, Meehan
and Radwinski could learn more about heatflow, and so on.

... our home was custom built by the previous owner, and has a very unique
open floor plan on the upstairs level, which constitutes about 2600 sq feet
... The downstairs level accounts for approximately 600 square feet

.... 255K Btu/h for 3200 ft^2? Does this house have walls? :-)
You probably need a blower door test and more insulation.

The entire length of the house, on both levels, has floor to ceiling
windows facing south that provide a breathtaking panoramic view of the bay.

Lovely :-)

... they are older windows, with metal frames, and are very inefficient --

That's OK in CA. A square foot of R1 south window with 90% solar transmission
would gain 0.9x1050 = 945 Btu on an average January day. On a 24-hour living
space, it might lose 24h(65-48.7)1ft^2/R1 = 391 with a net gain of 554. On
a low-thermal mass sunspace, it might lose 6h(70-52) = 108, netting 837.

the house loses a lot of heat when it is cold through those windows, yet
when the sun is out the vast southern exposure beams through the house,
heating it sometimes to the point where I literally feel like a dog locked
in a car on a hot summer day.

Congratulations! You have a solar-heated house! It needs more thermal mass,
maybe upstairs or below the ceiling, with an insulated wall between all
those windows and the 24-hour living space and airflow between the new
"sunspace" and the living space that stops at night.

... it can get windy, meaning it can get super cold at night. Yet when
the sun is shining, it actually heats up to the point that by mid-afternoon
I'm opening windows because it's too warm -- even in November (although
once the rainy season starts it will be cold all the time -- I know this
from our first month in the house, last February...

You might "open windows" with a thermostat and an exhaust fan...

NREL says 830 Btu/ft^2 of sun falls on a south wall on an average 52.2 F
February day in San Francisco with a 60.8 daily max. A square foot of R1
south window would gain 0.9x830 = 747 Btu and lose 24h(65-52.2)1ft^2/R1
= 307, for a net gain of 440. On a low-thermal mass sunspace, it might
lose 6h(70-56.5) = 81, for a net evil gain of 666 on an average Feb day.

I am looking to cut costs b/c I'm pretty broke right now, but not at
the expense of safety, comfort level, or decreasing the value of the
house by not getting the right furnace installed. Meanwhile we have no
heat, and I've no idea which (if any) of these contractors I can trust.

My advice: forswear thy fossil fuels and improve the solar performance.
It's easy, with high-school physics and simple arithmetic.

Nick