There are (at least) two different types of stains for concrete. One is a surface coating, the other is acid etching.
The former can work, but isn't as durable/lasting. I believe most of the products you get at the big-box stores are more of a top-coat type of stain.
Acid etching is just that, it's a stain containing an acid that actually etches the concrete. This is the typical type of stain you see in home magazines and such. The etching provides a deeper stain that is more durable and not prone to flaking.
I've done the latter, and applying it is super-easy. It's the prep work that can be a chore--namely if you have adhesive on the concrete already. In our case, we has tile mastic, which we had to dissolve using soybean based stripper. It worked great but was incredibly messy.
If you don't have glue to deal with, however, you should be good to go. Order up the stain (lots of places online to get it). You then dilute it per instructions and spray it on the concrete. I used a garden sprayer for this. You then let it sit for a period of time, and then you apply a neutralizer to stop the reaction.
After that, you apply water and mop/squeegee/wet-vac it up.
Then, you apply a sealer of some sorts. Common options are wax-based, polyurethane, and epoxy. I went with poly as it was relatively easy to apply (roll it on). That said, I don't think it was ever truly waterproof. For your needs, you may want to go the epoxy route.
(I'm assuming that you are in the UK).
Concrete floors without DPM are not unusual in houses of that age, so there is a possibility that it's not a cowboy job. The broken bitumen you mention was probably the remnants of an asphalt DPM over the slab. The wood has (as you suggest) likely been used to level the concrete. If they were previously joists for a suspended floor, some have certainly been removed, as the spacing is way to wide for floor joists.
Without ripping the floor up, I can see two basic options:
1: Fill the gaps with concrete and get an asphalt company to lay an asphalt screed over it to restore the DPM.
2: Fill the gaps with concrete, lay a DPM over it, insulation, then a concrete / fibre screed. Note though that this will raise the floor level by 75-100mm (depending on the insulation).
In terms of concrete, as it looks like it's not a structural element, you could just use a "basic" 4:2:1 concrete mix (4 parts stone, 2 parts concreting sand, 1 part OPC). Add enough water to make it workable, but not so that it gets sloppy.
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Frankly, Stacey, the thought of a concrete-over-wood floor frightens me badly. IF you can determine that your existing framing can carry that much additional "dead load", then definitely rebar and remesh (the mesh) well or it'll crack. You should know that the floor will want at LEAST 2" of thickness, which means each square foot of flooring will weigh at least 25 pounds, so the floor for a 12'x12' room would weigh a total of 3600 pounds (nearly two tons).
Talk with your local concrete companies - they may be able to mix a batch of lightweight concrete for your purposes - if you're dead-set on this. It WOULDN'T hurt to employ an engineer to verify that your structure will tolerate that much extra loading, though.
You do know that the concrete floor will be very cold underfoot, right?
EDIT: ...and that it'll always generate "concrete dust"?