Day 1: When you get to Surigao, you can stay at Tavern Hotel or EY so you're just right in the city. You can just walk around if you just want to see the small city (you won't get lost -- it's so small) but if you're up for it, you can got to Mabua beach, the pebble beach, which is about 30 minutes away. Best to go at sunset; the sunset is nice there. Ask your hotel for transportation services to Mabua.
If you just feel like crashing in your hotel, stay at Le Chard bed and breakfast. Gaisano mall is just walking distance so you can just go there for your food and other needs.
To eat, I suggest Ocean Bounties near the palengke, Islands right in the city, Calda Pizza along the boulevard, Bastie's right outside the airport or Mooon Cafe in Gaisano mall.
If you want to try the nightlife, you can go to J-Spot next to Calda along the boulevard for house music and drinks or Kody's for a band and beers.
Day 2: I don't know about the boat rates and the only accommodation I know there is Club Tara. Pricey and not well-maintained but you don't have a lot of options.
Day 3: Yes, you take a boat from Socorro to Dapa in Siargao from Bucas Grande/Sohoton. I'm not sure of the rates either but yes, make prior arrangements.
When you get to Dapa, you can take a tricycle or a van to General Luna. But if you want to get around GL, I'm sorry but your only option is to ride a motorcycle or habal-habal. That's how people get around in GL.
Lots of accommodations. If you want somewhere near Cloud 9, try Ocean 101, Cloud 9 resort or Sagana if you can afford it. If not, there are so many there, you can just literally walk around and go door-to-door.
Day 4-5: It depends on what time your flight is. If you have the 9:35AM Cebu Pacific flight, you have no choice but to leave on Day 4. If you'll be on the 1:20PM AirPhil flight, you can leave early in the morning on Day 5.
Travel time is between 2.5 to 3.5 hours by boat.