Sounds like you have a great trip planned!
"From what I’ve read, 15% to 20% is expected for an “acceptable” level of service, more for ‘great’ service. Is my understanding correct?"
Yes.
"More critically, who do I tip? Waiters, taxi-drivers and bartenders are obvious ones. What about ordinary shop attendants? Dry-cleaners? Fast food restaurants? Car hire? Bus drivers? Hotels? Do these businesses expect a similar level of reward?"
Ordinary shop attendants, dry-cleaners, fast food joints, bus drivers (unless said driver is conducting a tour) - No tipping needed. You’ll see fast food places with tip cups by the register, but IMHO unless you’ve given the counter person a complicated order there isn’t any need for a tip.
Car hire - if you mean a taxi cab rather than a rental, you do tip. I usually round up the fare to tack on about 15-20%, plus $1 per bag if the driver is handling luggage. Car rentals, like from Hertz - none needed.
Hotels - If you’re staying in a place with bellhops, etc., then definitely tip the guy who brings your bags to your room. I usually tip these guys $2 a bag, more if they are especially helpful in some way. (I once tipped a guy $10 for getting me a better room when the one I was slated for should have been taken out of service for repairs.) If a doorman gets you a cab, $2-5 is fine (depending on how swank the place and how generous you’re feeling.) If you stay in a hotel for a couple nights or more, $2-3/night left on the bureau for the maid service is good form.
"How does tipping work with change? Say I get my hair cut for $20. Adding a 20% tip, I’d like to pay $24 – but I only have a $50 note. How do I convey the amount of the tip to my hairdresser so I can get $26 change? Do people just come out and say what they’re leaving on top of the bill?"
If you’re in a place where someone like a hairdresser could reasonably expect a tip, they will likely give you your change of a $50 in a series of bills that would make it easier for you to leave a tip, e.g., $30 might be given back to you as a $20, a $5 and five $1 bills. If not, you can simply ask them to give you change of a $10 bill (or whatever size bill you need to break).
"How do I tip people when there’s no base amount of money involved in the transaction? – like cleaners and valet parking attendants. How much money do these people get?"
I have personally never tipped a cleaner, but then I seldom have things dry-cleaned. Parking lot attendants, I tip like bellhops - a couple of bucks is fine, usually, unless you’re in a really swank place and have asked someone to look after your Lamborghini. 
Please note that at the airport, the baggage handlers at the curb (who get your bags in from the curb to the check-in desk) will also expect something. $1 or $2 a bag is fine.
"In Australia, the correct etiquette is to shove money in the direction of your bartender when they serve your drink. They take it and return the correct change. In the US, how does this work with tipping? Since most of the time it’s too noisy to hear the price of drinks, how do you know how much to add on to the price of a drink? Also, the problem with having incorrect change, as above."
If you’re in a noisy, crowded place, hand the money to the bartender directly and wait for change back to see how much it’s cost (you can try shouting to ask the price, but I find that seldom works). Figure somewhere around $5 per cocktail, a bit less for a beer. Leave your tip on the bar, shoved toward the bartender’s side - they’ll pick it up when they get the chance. If you’re someplace quieter and in the midst of conversation with your friends, you can just put put your money down on the bar after you give your order. The bartender will tell you if it’s not enough; otherwise, he/she will just take your money and put the change back next to you. For a few rounds of drinks, it’s not uncommon to put, say, a $50 down and let the bartender keep tapping that money until you’re done. Tip money should again be pushed toward the bartender’s side, so they know it’s meant for them. If you need change, you can just ask for it - believe me, they won’t mind!
"What’s a ‘take a penny- leave a penny’ dish?"
Australia did a wise thing in abolishing their $.01 coin. We haven’t yet, and sometimes pennies can accumulate like rabbits (it seems) in your wallet. The “take a penny - leave a penny dish” is meant for people who don’t want the pennies in their change, as a goodwill gesture towards the next poor sap who has a bill of, say, $2.36 and no pennies of his own. Said poor sap can take a penny from the dish and thus avoid getting back four pennies he doesn’t want either. 
"(From a recent Pit thread) How does one respond to, “wassup?”
"
“Not much, man, how 'bout with you?” 
Have fun!