I have 2 questions about the plot:
1) I have just finished the quest with the assassin who killed the duke and it makes no sense to me. What was the point of letting the assassin guildmaster go only to trap a mere hired killer? I thought that the most important goal of this quest was to exposethe duke’s son who commissioned the murder and only the guildmaster could testify against him. All the guards in the castle were talking about wanting to cut his throat, they only wanted to be sure it was him who had the old duke killed. Instead, the commander of the guards is satisfied with killing just the one who carried out the murder and forgets completely about the one who commissioned it. Why is that? What do you think?
2) I have just read about the ending of the game (I know it isn’t right to do so without having finished the game yet, but I was too curious) and I read there is a scene where the hero is tempted to kill a child with the spirit of a demon inside but at the end doesn’t do it and after that he is seen to carry the child away. I would like to know if this child is supposed to be the main “bad guy” in Divine Divinity 2 or is it another character? Does the ending of Divine Divinity 1 stay open or is it explained in part 2? I am asking that just because I hate open endings and would like to be prepared if this game has one.

If you are currently playing DD, the ending is discussed in detail in this thread. If you don't want to know that info you should stop reading here. . . =)

*** Spoiler about some of Divinity 2 ***
Divinity 2 : Ego Draconis sort of continues the plot. It was released quite recently, so I doubt we'll see it on GOG any time soon. However it's not a bad game, it's fun being able to turn into a dragon, and it is a nice progression into a 3D environment. Anyway, as far as the plot goes, it continues about 30 years later, there are books around that fill you in on what happened in between, but suffice to say - you don't play as The Divine One.
Oh, and I'm afraid the ending to Div 2 is rubbish. Don't get your hopes up.

I know. But they are planning to release an add-on this year. Maybe the ending there will be better.

I just read this, but then I watched the ending sequence on youtube and it's completely different. While the thread suggests he killed the child, the ending on youtube leaves the question open. Why are they so different? Did the author of the thread make something up?
Did the creators of the game ever explain if he killed the child or not?
And what about the game Beyond Divinity? I read somewhere that in this game you find out that the main villain of Ego Draconis is the child the player didn't kill in Divine Divinity 1. Is it true?

I try not to post any more spoilers than necessary to move game play forward for someone. Don't want to spoil it for others.
If you Google for DD, you can find lots of info on the plot and ending . . . =)

OK, if you will insist. DO NOT read on if you do not want spoilers for DD, BD and Div2.
The Divine takes the baby, and attempts to raise it as his own, believing that the capacity for good or evil is inherent in all beings. He does a pretty good job of it, but the black ring infiltrate his household, and the child (now a teen) falls in love with their agent. She slowly corrupts him, until the Divine one discovers her identity, and has her executed. At this point the child (called Damien) bursts into the room at the moment of her death, and soulbinds himself to her. The intracacies of this are examined in Div2, but after this he leaves the Divine One, joins the black ring, and begins his war. He is then known as "The Damned One".
EDIT: - and yes - your "enemy" in Div2 is Damien. Though there is yet another winding story. He is the Damned One though - so there's a hint as to his nature.
In BD, as far as I've got, you see statues around the place of The Damned One, I don't know how much info is revealed as to his history, but he is there. I don't think that you contribute to the DD storyline in BD (haven't finished it yet).
In Div2 this has all happened, and you begin the game in Rivellon about 30 years later, the Divine one is dead, killed in the second of two wars with Damien, betrayed by a Dragon Knight.
None of this is really a spoiler to be honest, as you are able to find all this out really early on in Div2. In fact they try really hard to get you to learn it all.

Another question: is there any hint in the second game about what happened to the elf woman the hero fell in love with in the first game? Did they raise the child together or did the hero do this alone? And did she get killed in the war as well or not?

I honestly couldn't tell you for certain, but I didn't notice anything about that. The thing is, you aren't playing as the same character in Div 2, so you only really acquire history of events if you find a book that tells you about it. Div 2 is more about developing the Dragon Knight character, and getting you up to speed on that whole side of things.
If you're that interested, buy it. It's not on GOG, but it's not a new release, and shouldn't be too pricey. I enjoyed it personally, I'm looking forward to the expansion, so that, as you said, the ending can actually be decent (reading up on the expansion, I think we might get a bit of closure there).