-- In layman's terms, what is a "child chain" and what exactly does it do ... beyond being a shorthand for the real thing and, consequently, faster in the transmission/recording of data. Specifically, give me three examples if you will, practical ones, of real-world use of a "child chain".
I'm no expert, but I will try it:
Child chains are comparable to the currencies of the actual Monetary System. They can be used, for example, for bonus points for companies, gift-card systems, steem-like content monetization systems, or in-game currencies.
But they have two very important advantages: All present "monetary system" solutions require transaction fees to be paid in the base currency (e.g. NXT or NEM). With child chains, the fees are paid by a so-called "bundler", who adds the childchain transactions to the main chain. So participants of these currencies can pay fees in their child currency, or not pay any fees at all (e.g. if the bundler is the issuer company and decides to pay the Ardor fees without demanding a fee in the childchain token).
The second, and even larger, advantage is that child chain transactions can be pruned, so most nodes must only save the lightweight Ardor main blockchain.
-- I understand that users of these "child chains" will have to pay for them. To who? and through/depending on what?
Child chain issuers probably will have to pay a "issuance fee" like the creators of actual "Monetary system currencies". Child chain users normally will have to pay transaction fees in the child chain currency to the "bundler" (see above) so the bundler can benefit from it. This creates an incentive to actually do the "bundling" work, which does not only waste resources but also fees in the Ardor currency. But as I said above, I think that companies can opt to not require fees from users if the child currency benefits them, e.g. from a marketing point of view (in the case of bonus points, in-game currencies from commercial games ...).
-- In what way, if any, are those "new" features listed in the OP, any different from the currently widely available so-called "smart contracts" and "decentralized markets"?
That they are limited and mostly hard-coded, so the possibility of bugs like the DAO desaster is much lower (almost inexistent) because they can be tested much more thoroughly and are much less complex. It's actually a system that puts less emphasis in flexibility, but more on security, more bitcoin-like than ethereum-like.
But why is my current balance only .08% of my expected balance? Can anyone explain this?