For live music, definitely try to catch Aldrine and Ryan! Unfortunately, I discovered UU about a week
after leaving Kaua'i. Shaka Tacos in Kapa'a is a fun spot for music. Damn good Mexican food, too. Pretty much all of the touristy resorts will have some kind of music. On a good day, you might catch some locals jamming on Kealia Beach.
For uke-buying, Larry's is awesome. Really nice people, and a great selection. Their flagship store is in Kapa'a, and they have a small store in Koloa too. Last time I was on the island, the Koloa store had literally just opened (the counter was still a cardboard box). WickedWahine's post suggests that they changed the Koloa store to specialize in the lower-end instruments sometime since. That same week, Bounty Music in Kapa'a (where I bought my Kamaka) became Kaua'i Music and Sound, and I'm guessing they're still trying to make the transition and acquire some new brands.
Never been to Scotty's (it was closed literally every time I passed it), but I've heard good things.
There's a store in the shopping center in Hanalei that carries Mele, as well.
Most of the good gift-shopping is in Koloa, Kapa'a (ughghghggh traffic), and Hanalei. There are touristy souvenir shops all over the island, but they all sell the same stuff. More fun to buy stuff made on the island.
I second the puka dog recommendation. Expensive, but soooooo worth it. Also, Jojo's Shave Ice in Waimea, and Tomkat's Grill in Koloa. For super-cheap eats, Aldrine is right: Sueoka's snack shop in Koloa is the best.
Shoot the admins a PM or email to find out how to meet up with them. I think the "office" is Ryan's house.
Have a great time on your trip!