Just read over this, TL;DR warning I guess!
Just to note from the start, you won't get a complete cosmetic facelift with any method because the rust is breaking through nickel (?) plating on some bars, but you'll get a smoother, rust-free surface that will give the clamps smoother operation so doing a little work on these is well worth the effort.
On the smaller clamps which have no wood there's the possibility of using a rust soak to remove the rust, by immersing the entire clamp. Personally I would consider this overkill for clamps, but everyone has their own standards and it would be effective and thorough (reaching areas you can't even see, much less access by other means).
My favourite soak is molasses and water for a number of reasons (very safe, non-toxic, can be very cheap, doesn't seem to harm good metal even if parts are soaked for extended periods of time). Its only disadvantage is that it's relatively slow.
Note: Evapo-Rust is reputed to use similar chemistry to molasses solutions. I can't confirm this, but they work very similarly and apparently they smell quite similar.
Electrolysis could be used in a similar way. This shouldn't affect paint that's in good shape and firmly adhered to sound metal, but if there's hidden rust under any paint it will flake tend to flake it off.
Edit: I would be hesitant to use either method above on these smaller clamps however if there are enclosed springs as you mention in a Comment. It's nearly impossible to dry these out thoroughly enough after soaking in watery solutions so that more rust wont be encouraged in future, even after drying with a hairdryer and drowning the area with WD-40.
Because these are just clamps this would be the method I'd most likely use myself. Also, because you can't soak the long ones due to the wooden handles it means you can use the same method for all clamps, probably ensuring the most consistent results.
This is also partly because the rust is so minor, heavier rust I think should always be tackled with chemical or electrical means first before any abrasion is used to finish off.
It's likely to also be the fastest method, since you can have de-rusted one or two clamps before you'd even made up a soaking bath or an electrolysis tank!
You might find that rubbing down with coarser steel wool (0 or 00 would probably work best) lubricated with mineral spirits or WD-40 will do everything you need here.
I wouldn't use Scotch-Brite or similar non-woven nylon pads sold as "synthetic steel wool" as a substitute for steel wool here. The abrasive action is not at all the same and the nylon pads will scratch any remaining plating more than steel wool. Steel wool has more of a scouring action while Scotch-Brite has a sort of sanding action (because it has small abrasive particles in the web).
After removing as much loose rust as you can you're pretty much done. Because you used an oily lubricant the exposed steel won't be prone to re-rusting immediately (as it can be with chemical or electrical methods) because the metal has oil impregnated in the 'pores'.
But you need to apply something more for long-term protection from further rusting. Obviously you can use whatever your favourite anti-rust treatment is but I would highly recommend paste wax here as it provides better protection on things that are handled frequently (who doesn't grip F-clamps by the bars when moving them around?) and it improves slip, without the surface being greasy. Win win.
My first instinct is to take the clamps apart, but they aren't built to be serviced. They seem to have some sort of rivet at the bottom of the bar that prevents the movable jaw from falling off.
On some clamps with long bars of this type this is a roll pin, which can simply be knocked out. With rivets they can be drilled out (to be replaced with fresh rivets later) if you really wanted to get the heads off, but it's perfectly doable to service the clamps fully assembled.
But I'm not sure if that would mess up the finish of the black bars?
If you decide it's necessary you can get proprietary colouring solutions to give metal a surface colour, similar to the treatment used in the factory to make these dark initially.
The dark finish is not merely cosmetic in case you don't know, usually it provides some rustproofing as well.
For the larger clamp, there are serrations on the bar, and I am not sure what I should do to remove the rust on that.
This is another reason not to use conventional abrasives, sandpaper especially. If you find the steel wool won't do enough you could switch to using a wire wheel in your power drill or drill press, but a hand-held wire brush might be sufficient (it often is for light rust). In case it's not obvious best to brush across the bar not along it.
My primary concern is the rust, but sometimes I go overboard on projects like this, and so I'm also wondering what kind of paint would be appropriate to touch up the jaws. I'm not sure I want to go that far.
Clamps are some of the real workhorses in the shop, they get used a lot, they can't be babied when your focus is on keeping the wood from coming to harm, they sometimes get dropped. So I think it's realistic to expect them to get banged up a bit over time and to look like it. However saying that I don't think it's going overboard to repaint tools sometimes. This isn't only about making them look nicer, paint is also a much more permanent rust preventative so it's not merely a cosmetic thing.
The paint type for this kind of thing would be an enamel. I can't give a specific recommendation but in general you'll get a much tougher paint if you go with a traditional enamel (brushes cleaned with spirits) than with a waterbased enamel. The latter have come a long way and they're not bad any longer by any means, but they're not generally as tough.
Also, is it possible to buy replacement pads for the jaws, or am I out of luck on that?
Many plastic pads have a slight texture in them to help with grip and this can be impressed even into some hardwoods (and very much so into softwoods) so I think it's better to affix plywood, hardboard or MDF pads to clamps, which provide good grip but no risk of transferring a pattern to the workpiece (also lessens the need for intermediate blocks of wood).
Normal PVA-type wood glue will work to attach the pads even though it's not for metal or painted surfaces, the grip is good enough to keep the pads in place but not so strong (as with epoxy) that replacement becomes difficult when needed.