We all are waiting. Here's my theory on your questions:

1. She is his sister - either younger or older. Remember back in those days powerful general's daughter or siblings married into the royal family.
2. I think she either has to be in love with him, or he has to be in love with her, or they mutually have to be in love with each other in order for her to see the sword.
3. He became the Goblin because 'God' wanted him to have his revenge for his wrongful death but also teach him about love - the kind you sacrifice yourself for. Remember, in his 300+ years of living, he never had a bride or someone he loved. He only fought battles and killed other people's loved one's. So it is kinda of a punishment and a lesson.

The first question was what I wondering too!
And I think @Daisy01 's answer makes sense.

I have another question actually...
I don't understand why did the goblin (shin) got executed as traitor... What did he do? Or what did he accused of?
Or is it just a super random execution just because the king is jealous of him, so without any reason at all, he just claim shin is a traitor?
I watched the scenes 2 times and read the recap at dramabeans but I still don't get it...

@soracantabile, because it is a crime to place yourself above the king. Remember the king's adviser's order: there is God - King - General - People. But in his mind (the King) he saw: God - General - King - People. It was basically a badly patched up accusation of him (the General) trying to overthrow the crown.  

@soracantile, any general who hears crowds cheering for him is as good as dead in kdrama land. Add a Grima Wormtongue behind King's arm and we can only wonder how can he managed to make it so far still holding his sword.
Regarding point 3: it wasn't the general who became a goblin, it was his sword. See the first scene. They're played by the same actor, so it can be confusing.
EDIT. After ep 3, it turns out he was a general in his past life afterall. But I still think intro lines are pointing more to his sword than him.

When a man dies, what's left is a ghost. The process of creating a goblin is different and explained by the intro lines (object + intense sentiments). The sword was charged with all those tangled deaths, anger and need for revenge and somehow transformed into a goblin. I guess it carries some reflex of his master personality / sentiments. It has to atone for the deaths it brought (hence immortality and endless pain). It tried to revenge his master.
Actually I don't think it's very clearly differentianted in the show. Main hero is just not a ghost, for the sake of diversity I suppose. (He should also eat liver and not steak and probably be less human looking and uglier, but I doubt anyone will complain here).
In Arang and the Magistrate /possible spoilers/ goblin was created by means of the dark magic, by mixing and compressing several human souls. He had no resamblance to, uhm, source material. That was one of the scaries shits I saw in a kdrama so far, Jack Nicolson peeking out of the door level of scary.

I'm wondering about the green ring too. The young queen (played by Kim So Hyun) first had it and then Eun Tak's mom bought it from the old granny (who turned young again) And on one of the trailers, I saw that Sunny (Yoo In Na) has it. 

Any theories about this? 

I like that theory about the old lady! I thought (since apparently she is very old and can change her appearance) she was some kind of witch, but your theory is more interesting :P
Also, about the grim reaper, I think he said that he uses his hat (wich the goblin continuously criticize) because when he is wearing it, humans cannot see him.