I composed my initial Answer based on how the Q was initially posted, as it has been revised this will feature a blend of quotes from the original and the current wording..... because I CBA to completely rewrite ^_^
If I don't have any plywood available, what should I factors should I consider when selecting other wood to make my cleats from?
This embodies the belief that seems widespread that plywood is actually the most suitable material. It's not. I actually tackled this in my initial wording and this part I can adapt to here. Because of the orientation of the plies, and the known weakness of some plywood (weaker glue joints, filled knot holes, frequent voids etc.) one of the most common choices of material is arguably the least suitable; even with quality stuff the loading is inherently trying to split the plies. But I'm not actually arguing against its use because failures seem to virtually never happen despite this, which is a testament to the inherent stability of the concept. And a pointer to the heart of my Answer.
The material doesn't have to be that strong
I really meant it when I said in my Comment that you can use just about anything. As I go on to say, most French cleats are hugely over-engineered for their actual loads; in fact any reasonable load that might be considered and without fastening exclusively into studs or a solid wall, but the holding power of drywall anchors/rawlplugs is a topic best left to the Home Improvement SE :-)
So really almost anything? Yes, including lesser hardwoods like poplar and most softwoods, even cheap crappy stuff1, and yes, also MDF2.
what properties (other than straight & flat) should one look for in the scrap lumber pile?
To be perfectly honest I'm not sure that straight, and particularly flat, is important. This sounds wrong but let me explain:
you will naturally take out most wind or cupping because of the way the strips are fastened to the wall and to the back of any cabinet or tool holder being hung up1;
with stock that's not straight, as you know the shorter the lengths you cut a bowed piece into the more you lessen the curvature. We take advantage of this all the time. So wood that's not straight could still be utilised for the (typically quite short) strips attached to the backs of things being hung up, with minimal loss of material.
Other than the aesthetics this is perhaps the ideal opportunity to use up material that is not suitable for many others uses. Including stuff that has knots in unfortunate locations that you might worry about in other contexts — because any significant load tends to be spread over a wide area a weakness in one spot on the cleat fastened to the wall becomes essentially irrelevant. There is no load applied just to that weak spot, unless by chance or planning a particularly heavy and narrow item is hung at that exact location!
Is a softwood, like SPF (framing lumber) considered strong enough?
That's a good question because from what I've seen in discussions in woodworking circles the answer would generally be no. However, some of that is just bias against using construction wood for anything that 'matters' and while based on legitimate concerns about the quality of so much of this material worrying about it being too weak for this purpose seems groundless. In the UK and Ireland it is very common to see softwoods employed for cleats in kitchen installs and for shop cabinetry and tool storage, and the success of these3 is a testament to softwood being perfectly adequate.
Dimensions aren't critical
In addition to the entire system not needing to be based on strips as wide as are commonly employed (3" and sometimes more is not uncommon if we look around online) the cleats on the items being hung up don't have to match those on the wall. Not infrequently I've seen a sort of blind assumption that the two cleats have to be the same — perhaps stemming from the very common method of ripping a width of plywood down the centre on the tablesaw to create them — but in fact the only thing that really matters is that the bevelled edges match.
And speaking of which...
Bevel doesn't need to be 45°
This is, by far, the most common angle chosen for the bevelled edge but some people feel there's a little too much scope for things to easily be lifted off by an accidental knock or by working their way up the slope for front-heavy items.
That tendency for some kinds of projecting tool holders to tip forward and begin to ride up the bevel can be lessened or prevented entirely using even small brackets underneath if needed, or by something more elaborate if needed.
Whether the risk is real or not for your application if that is a concern just go to 60° and you end up with a much more secure footing. Note that because the edge ends up so much thinner you might want to round over or flatten it rather than leave it sharp.
Very high loads can be spread over more than one cleat
TBH I'm not sure if this is necessary, but if you need to do it for peace of mind why not?
Elephant in the room: the chief potential sources of failure with French cleats are the fasteners or the wall/cabinet material, not the cleats themselves.
1 This is not based on theory, it's based on seeing French cleats made from cheap crappy stuff that are having no difficulty whatsoever in surviving year after year in use with heavy cabinets hanging from them. This is both in a workshop environment and in kitchens.
2 See post by Les Mahon here on UK Workshop. The Benchmark site that he links to is actually one of the things I saw early on that gave me confidence in the strength of the system can be if implemented well, and this has been amply reinforced numerous times since then. This includes seeing cleats made from thinner and narrower stock (firsthand, as thin as 10mm / ~10/32" and only 30mm / ~1.2" wide, and I've read accounts of cleats 1/4" thick only 1" wide) that hold just fine.
3 Let me be clear here, the apparent complete lack of failures in practice.