Billy, you're not alone in wanting to know "the right answer" for a question- I'd love to be able to give it.
What struck me immediately here though was the missing component- Billy, basically.
What kind of cooking do you do? What kinds do you LIKE to do? Are you allergic to woodsmoke? Do you enjoy cutting, splitting, handling firewood, or are you really too busy? How much "extra" time do you have- either to wrangle wood, or propane containers?-
Etc. Hopefully you get the idea. Who you are- what you need- and even what makes you happy- all these considerations are genuinely IMPORTANT to the answer.
You are important. We need to remember that.
"Sustainable" practices WON'T be- if they make people miserable, and they won't stick to them.
Which seems obvious, but quite a few enthusiasts will, in the excitement of the moment, adopt practices that they can't/won't - uh, sustain. Because in their enthusiasm for the greater good, and the benefit to the planet, they forgot- WE are part of the planet we're trying to save here- and we matter, too.
The whole decision- what kind of fuel SHOULD I cook with - can get pretty crazy complex if you keep picking at it.
Propane is a fossil fuel- bad carbon. It's mostly delivered on trucks- diesel fuel; more fossil carbon. Where does your local propane actually come from? Natural gas is often moved in pipelines/pipes- pretty efficient, if available- but still fossil carbon.
Electricity is mostly coal (bad), and nuclear (BAD); with minor bits of natural gas (badish) wind (ok) and hydro (okish) - depending on where you live. If you've got the option as some do to essentially purchase straight renewable electricity- that could make a difference in your decision.
Wood is "current budget" carbon- good carbon; and it CAN be renewable, though like everything else wood can be done badly. If you live in a city - it may not be legal- most available wood-burning stoves are much dirtier than they have to be, and wood smoke is pretty irritating for the neighbors. Do you have a good supply? The space to store it, the time?
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As an aside here- firewood is kind of dominating my life at the moment- because of the floods last fall, and global warming. I cut and gather it myself; the floods made harvest much more difficult/much more time consuming- so I wasn't able to do my normal autumn wood cutting.
And- the firewood we had cut from our own plantings; stacked, curing/drying - got soaked thoroughly by the 14" of rain in August/Sept- and is unburnable. Given normalish weather- it wouldn't be nearly so wet, and we'd have had days of low-humidity sunny windy weather in Sept/Oct that would have dried it very well. So in fact I'm cutting firewood every other day- and burning it fast, since it's cold this winter; lots of below 0 F nights.
There are a LOT of other things I need to be doing- but here I am. The Little House has no backup heating system- it's firewood only (with a little passive solar boost- not useful at 1 AM).
Will rainy autumns happen more often? Don't know. This wet autumn, though, may be the thing that pushes me over the edge into adding a layer to my firewood process- a drying/storage shed.
There have been many years where a rainy week in November got my wood a little wet- making me aware that if all the winter stacks had been under a roof, I'd be burning less wood; doing less hauling- but- it's always been a fairly minor factor. And every time that happened, I've done mental calculations- what would it cost me- money, time, and new habits- to design and build a wood drying shed? A bunch. How big would the benefits be? Considerable. Balance? Kind of six of one, half a dozen of the other.
This year is the first where all the stacked wood is so wet it's nearly useless. I can make it burn, but it gives little heat, and clogs the chimney fast. The balance may have shifted- instead of being a minor improvement, the shed may now be a necessity, up-front costs or not.
It strikes me that this kind of shift may be another major aspect to global warming- tiny local processes/technologies may no longer be reliable. Pushing people over all kinds of edges.
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Maybe the best I can do for an answer to Billy M's very sensible question is to describe my own answers. More than one answer, since I've changed, over time.
When I first moved to the Little House, a major factor in the calculation was money- we didn't have any. We DID have wood- 40 acres of hardwoods.
With that in mind, I designed the Little House to use wood both for heating and cooking- all year. Including our sultry hot continental summers. (It's a huge advantage to be able to design a dwelling from the ground up- with all the integration factors being considered. I still missed a few, of course.)
The House can essentially be tweaked to function like a big chimney/cooling tower in the summer - the downstairs has big windows in all 4 walls; the upstairs/loft has one huge window (floor to ceiling) on the north, and a normalish window on the south. All the windows but the small one upstairs open on a hinge- so unlike a sash-window, where the actual opening can only equal half the window area at best, the hinged windows when open make holes equal to the entire window area- huge, in our case.
And- there's a BIG opening between upstairs and down- so if all the windows are open, any heat from the stove is quite free to rapidly move up, and out.
It works fine, too- we did all our cooking with wood for probably the first 5 years or so.
Then several things changed- we had children (available time and energy vanished), we got involved in other projects that were important too; and we got a little money coming in.
Suddenly it became more sensible to use propane for cooking in the summer.
And that's what we still do. The stove that heats the house is a modern Canadian stove designed for both heating and cooking. If we need heat- it's on, and we cook with wood. If we don't need heat- we cook with propane. The time required for the propane is a small fraction of time needed to cook with wood in the summer- and no question, July and August are a little more comfy if we don't have to crank up the woodstove to make a cup of coffee, or soup for lunch.
One departure from that practice can be canning- if we're canning tomatoes or whatnot- we will usually use wood- canning takes a lot of heat; and ergo a lot of money.
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One more aside- cooking in China. As part of my work, I've hiked well up into the mountains in a number of places there, out into nearly untouched countryside. These are ancient communities, long in "balance" with their environment. Chances are, this clan has lived here since these people were Homo erectus, not Homo sapiens. That long.
They long ago hit the limits of their environment; and adapted, in many ways. Only the rich can afford to burn wood- there's just not enough of it, and mostly it's needed for other uses, tools, furniture, housing. They burn- rice straw, and pine needles. Under a wok. That's exactly what a wok is for- cooking over a very quick, hot fire. Their whole cuisine is adapted in that direction- because of the primordial shortage of fuel.
I'll bet you could cook entirely on - junk mail. If you had the will, and someplace outdoors for the smoke to go away. :-) You'd need something like a ventilated 5 or 10 gallon steel can for the fire to burn it, and the wok to sit on- (I'm kidding- mostly... probably too many toxins in junk mail smoke to be good cooking fuel...)
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So. Answers to questions like this are going to be highly variable, I think. Forever. Because one of the most important components in the decision making algorithm is always going to be personal. How does this fit your life, your finances, your region?
If it makes you miserable- in the long run, it's not a good answer -
The good answers should leave you - solvent, not overworked, and satisfied.