Can a .44 be down-loaded to resemble a standard .38 round in recoil and use (so even less power than a .44 special)?
You can't realistically go much lower that published minimum .44Spl loads without a danger of squibs, particularly out of a carbine.
The felt recoil characteristics of .44Spl will never be quite the same as .38Spl because it uses more massive bullets. That said, both should have pretty negligible recoil from a carbine.
Is the price difference between hand-loading .44 and .357 significantly different? (I know that off the shelf .357/.38 is a cheaper combo.)
OK, in all seriousness, .44 typically costs roughly 30%-50% more per round if you purchase off-the-shelf bullets. The difference is potentially less if you cast your own, particularly if you have a free-to-cheap source of recyclable lead, but these are becoming fewer and farther between, and many handloaders don't want to expend the time or effort to cast their own bullets regardless.
Furthermore, new .44 brass costs significantly more, and recycled range brass is also more expensive because the supply is smaller; however, brass in both of these calibers usually has a very
long lifespan, particularly if the loads are mild.