1. Are there any "must see/do" things in any of these places?
Yes, but others will chime in before me at this point.
2. How common is Internet access, especially free wifi with lodging? I'm not sure what the practices are.
Variable - internet cafes are plentiful, but with your hotel, kind of like the US - the fancier the place, the less likely it will be free.
3. Is it crazy to bring 2 or 3 suitcases (for two people to share)?
1 per person plus your carry on - don't bring more. Many taxis are fairly small.
4. How accessible is laundry service? If I don't bring much luggage, I'll need to figure out how to get clean clothes.
If your clothes are a little dirty, so what. Most hotels have a laundry service - but you could pay dearly. A hotel can steer you to the closest lavanderia. Note that with the relatively short stays at each stop, if you choose this option, you need to hop on it as soon as you get somewhere.
5. Are there liquid restrictions on international or LAN flights? Meaning, can I buy a big bottle of shampoo, conditioner, mouth wash, etc in Lima and take it with me through the rest of the trip?
LAN has no restrictions as an airline. LIM won't let you through security with 100ml+ liquid bottles even though there are no restrictions flying i.e., to SCL or EZE. Besides, if you don't have your favo(u)rite shampoo or conditioner while on vacation, it is not the end of the world. In fact, it will make you a better traveler.
6. On that note, hotels in the US usually provide shampoo and conditioner. Is this common in SA, too, or do I need to make sure to have my own?
I think I've yet to not have shampoo in South America, and I've stayed in some relatively spartan (but not hostel) accommodations. If you miss a few days without conditioner, your frizzy hair will not ruin your trip. It will make you a better traveler.
7. Anything to make sure I bring from home that might surprise me? Like, should I be able to easily find Tums, Advil, feminine hygiene products, sunblock, or anything else like that?
Most everything is available - but may not be obvious, i.e., in Chile, you might need to ask at the counter for an antacid (even though no prescription is required.) You will likely not find your favorite name brand.
8. If you were going on a similar trip, how much USD cash would you carry? My preference would be to not rely on a bank/ATM in SA but it seems crazy to carry enough cash for 2.5 weeks.
Use your ATM, and see if you can find a local bank that will reimburse you for ATM fees. Also, get a credit card with no Forex % charges. Also note that smaller places don't take CCs or may give a cash discount.
9. I tend to take my laptop with me, but is this a bad idea? What about an iPad or netbook?
It can't hurt to have your laptop, esp. if it easily fits in your carry-on.
10. I have an iPhone on AT&T. What's the best way to go about getting a data/phone plan? Should I just get a prepaid phone? Is there one that would work across my whole trip?
You need to call AT&T to see if they have short term data plans. Verizon used to have a great one, but now they don't. A prepaid phone won't work in all the countries you are going to - you'd probably need different sim cards. Not worth the trouble. If you need to call home, there are many "locutorios" (with various names) where you can call the USA and most of the world, for cheap. Also, if you aren't paying for toll-free calls from a hotel, in Argentina they have cheap int'l phone calling cards - AR$10 (under $2.50) for an hour or so of calling the US. Many little kioscos sell them. If loved ones need to get in touch with you (i.e., real emergency) they can call your hotel. Otherwise, email is fine - enjoy your trip and don't worry about keeping in touch with North America each and every minute.