Solo Itself
I don't think there are any house rules required to play Pathfinder one-on-one. There are solo adventures for Pathfinder from Expeditious Retreat Press and have been for other D&D versions over time as well. All that's required is DMing. Since there's one PC, they will need difficulties tuned to them and some kind of safety net in case of bad luck - a NPC friend, or fate points, or something.
Investigation
If you are concentrating on more storytelling and whatnot and it's low combat then most of the problem isn't getting killed so much as getting stumped. If you're going to do lots of investigation consider Lorefinder, the Pathfinder/GUMSHOE mashup, which changes investigation skills to not block PCs. I playtested it and it brings the good parts of GUMSHOE (Robin Laws' investigative system that powers several Pelgrane Press games) to Pathfinder.
Advancement
In our Pathfinder games we stopped using XP a long time ago; the DM advances us when it's time. That removes the grind incentive and allows story development without artificial combat.
FATE
We also about half the time use FATE aspects and points in Pathfinder. Essentially we declare three aspects to our character, and when one gets us in trouble we get a point, you can spend a point for a reroll or a +4 if it can be linked to an aspect. FPs reset to 3 at each level. I am ambivalent about how much they really help with roleplaying, but they do present a consumable resource PCs can use when they think it's important.
Relationships
In terms of NPC relationships, I have a separate question open to get good relationship mechanics for Pathfinder but no one's come through yet really.
He just uses the +2 for the short sword melee attack.
That site has already calculated the attack bonus. As you calculated, the goblin is getting is +1 from BAB, +0 from STR, and +1 from size, for a total attack bonus of +2.
For comparison, note that he has a +4 with his shortbow: +1 from BAB, +2 from DEX, and +1 from size. (You use dexterity rather than strength for ranged attacks.)
slang definitions: BAB = base attack bonus, STR = strength, DEX = dexterity
Best Answer
Yes
Because of the nature of the rulebook, it includes things that are not part of the SRD. The website, while it contains a lot of the core rules, does not include all of the information that is laid out in the Core rulebook (Like the specific rules on how to make a character step-by-step, for example).
The d20pfsrd.com contains a vast wealth of information from the core books and is updated regularly, but I would not consider it a replacement for the core rules. It'd be like trying to play a FATE based game by just using the Spirit of The Century SRD.
Additionally, while d20pfsrd.com is very well maintained by a group of extremely dedicated volunteers, it occasionally has errors (which are typically corrected very quickly, but occur none-the-less).
Slightly off topic, but still relevant: By purchasing the Rulebook, you support Paizo, and thus make more core Pathfinder material possible.