"Regarding line, do tenkara fishermen carry spare line, or do I just cut my level line at home, tie my fly/tippet on while I'm fishing, and call it good? Is a spool of tippet on hand (and spare flies) recommended in case of snags, and if so how much?"
Getting started near home -
Level Line typically comes 25 yards +/- to a spool. Generally speaking - cut a piece equal to your rod length, and then add about 3 of tippet, and you are good to go. Anything longer than that becomes a challenge when landing fish. Later on you can go longer as you learn to "hand-line" the fish to a landing. I carry lines of different lengths that I hand-wind and them place in 2x3 ziplocks. Less bulk. No need for on-the-stream "cut to fit".
You can always add (tie on) more line or use a longer tippet if needed. Take the spool of line the first couple of times out for security. Definitely take the spool of tippet. You'll lose tippet to snags, the line is Much stronger.
Cruise the Tenkara USA site for opinions on level line versus traditional. I prefer level line.
"I'm taking a hard look at the Iwana rod 12' by Tenkara USA. They have a 60 day unconditional money-back guarantee that looks attractive"
The Iwana 12' is a great all around rod. TUSA has tremendous customer service and a web site full of critical and useful information. It is an excellent place to start.
"Wouldn't I want a longer line for lake fishing?"
Not necessarily. Depends on the lake, the wind, the brush, the bed profile, the fish, etc. I'm not a lake guy. But, Search: mihalik tenkara lakes. I've posted a couple of links here.
"Finally, do you have any suggestions on what flies to get and how many for, say, an overnight backpacking trip?"
Flys depend on many things. And in the Tenkara World (capital letters) there is an ongoing debate between the "traditionalists" and the "do what works" folks. For now go with traditional kebari and see what happens. They are amazingly effective. Try some simple #14 beadheads - like a Hare's Ear. Between the two, if there are fish, you will catch them. Most flies are intended to catch the fisherman. It's the simple well presented flys that catch the fish.