Some great answers so far, I'll add a few thoughts!
Hotel-wise: the Edgewater Hotel is nice-- it literally hangs over the water and is right near the Olympic Sculpture Park, which has a lot of great art and a killer view. You'll have a hard time finding a better sunset in Seattle.
Brunch at Boat Street Cafe is a good bet if you stay in this neighborhood (they are open 10:30 7 days a week).
The Walrus and the Carpenter is amazing seafood, but expect a wait on almost any night (Ballard is a decent neighborhood to wander around for a pre-dinner drink, tho).
Armandino's Salumi has AMAZING cured meat and porchetta sandwiches-- it's a weird place run by Armandino Batali (Mario's dad). Expect a line and they keep weird hours (great if you want an early or late lunch).
Crumble & Flake is the best bakery in the city (show up RIGHT when they open on the weekend at 9 for canele and kouign amann- they sell out quickly).
If you get good weather in August, grab the ($5, cash only, no change!) water taxi to West Seattle... Best skyline view of Seattle in the 12 minute ride over. Right on the other side is Marination Ma Kai (brick and mortar version of Marination, voted the best food truck in America).
Anything by Tom Douglas is worth a trip, so start your morning with Dahlia Bakery and then visit Espresso Vivace for a Cafe Nico - espresso, milk, cinnamon and orange. Victrola Coffee makes the best mocha in the city.
Tavolata is the best place to start if you want to explore the culinary scene. Their pastas are exceptional, and they can point you to the most innovative chefs in the city. Canlis is the heavyweight champion of Seattle cuisine; Shiro's has the best sushi. If you don't mind waiting in line, you'll have the best pork shoulder of your life at Paseo Caribbean Restaurant.
Tony, welcome to Seattle! It's a pretty great place, especially for coffee. I'll focus in that area since I know of some really great places worth checking out. I'll try to stay near the downtown area, since I'm not sure about your mobility.
If you're feeling super cheesy/touristy, or you've got some perfect snarky response in mind when people ask you why you went, it's kind of a must-do to stop by the Starbucks at Pike Place Market, because it's the 'Original.' The coffee's not great, but I feel like it's sorta a 'passport stamp' for coffee people, just to say they've done it. Also, it's not strictly coffee, but a stop at Top Pot Doughnuts will fuel you up with donuts and java before a day of adventuring.
I really enjoyed Citizen for coffee with food--they have some great sandwiches and are super close to the Seattle Center/EMP if you're walking around. In the same neighborhood (Lower Queen Anne), I've lately been haunting Caffe Ladro where they make a mean iced mocha and are happy for you to camp out for a few hours if you have work to do.
Hopefully this is a good starting point! You could probably stay super-caffeinated for days with this list, but I'll let you choose which ones are easiest to try.
Have a great trip!
The Ace Hotel is another good place to stay in Belltown (very central).
Coffee: I would argue that many of the places recommended by others in this thread for coffee are not going to appeal to you if you're into the new style of coffee that's been in vogue for the last few years (i.e., single origin, lighter roasts, fruity, complex flavors, etc.). To experience a more current, fresh, unique coffee, try these places: Trabant (I've only been to the pioneer square location), Stumptown Coffee (this is where I'd stamp my passport), and Slate Coffee Bar (the ultimate in new approaches to serving coffee--you'll see what I mean if you go). I also like Victrola Coffee.
Restaurants: I'd second Tony's restaurant recommendations and add Altura for excellent, fancy, very expensive Italian food on Capitol Hill, or Spinasse for a more country-style, merely expensive Italian meal (also on Capitol Hill--great neighborhood for food and nightlife). Other favorite restaurants (some of these have been mentioned by others): Westward Little Gull Grocery for the view and outdoor fire as much as the food, Sitka & Spruce, Armandino's Salumi, Bar Sajor, The Corson Building (single seating each night, small place--call ahead), Revel, Delancey (pizza), Rain Shadow Meats Squared (lunch/afternoons).
A few spots:
The Independent Pizzeria (reminds me of Delfina Pizzeria in SF). If it's warm, go and sit outside (it's a tiny spot in Madison Park): http://www.theindiepizzeria.com/
First off, let me start by saying that Seattle summers are to die for. Hopefully, the weather holds out for your trip.
Now, to answer your questions...
I agree that all of Tom Douglas' restaurants are worth visiting. My personal favorite for brunch is Brave Horse Tavern, where I usually get the pretzelwich. I mean, you can't go wrong with eggs and bacon on a pretzel bun.
Mamnoon is a great place for a unique middle eastern meal. I have never tasted anything as sweetly delicious as the rasberry sorbet.
If you'd like to have a good view and good Mexican eats, try Little Water Cantina. The margaritas are up there in my book.
Serafina Osteria & Enoteca is one of my all-time favorite restaurants. I had mussels there earlier this summer that transported me back to the San Juan Islands.
At Paseo Caribbean Restaurant there's always a wait (you can call ahead and order), but it is SO good. I get the prawns sandwich.
Take I-90 to the Alpental parking lot (exit 52) and hike 6.5 miles roundtrip to beautiful Snow Lake. The other day, we watched the sun go down behind the peaks at the top it was majestic.