1. Is the gospel a worldview? I would argue that the gospel is the story of Jesus' life, death and resurrection - in other words, that the reason we call first four books of the NT 'gospels' is that this is what they are! In that sense, the gospel is not a worldview. I think that's important - a worldview is made up of lots of other things beside story (see The New Testament and the People of God, at length!), but it is the story of Jesus that constitutes the proclamation of good news. Only in a story - rather than a general discussion of what the world is like - can we hear about something that has been accomplished for us, by someone else.
2. Is the gospel logical? Can it be presented logically? I suppose I'd want to recast this question. A worldview can be logical or not, can be presented logically or not - a story is judged on different criteria, primarily (if it purports to be a story about history) the criterion of reference: did this actually happen? Still, the gospel story presupposes and entails a worldview, so can we talk about that being logical? I think the gospel-worldview is logical if and only if the gospel story is true. Therefore, I think we must take people to the story of Jesus rather than to the worldview if we are to make a convincing argument.
3. Can we do natural theology? Is there such a thing as general revelation? Nein. 'Nuff said.
4. Does this entail a 'super-spiritual' way of looking at the gospel? Does it mean the gospel is something totally different from every other message in the world? No, and yes. Not 'super-spiritual', but certainly God entering his creation is a unique event which cannot be compared to or ranked alongside anything else! So can we compare the gospel-worldview to another worldview? In one sense, yes: we can see what each has to say on different topics. But in another sense, no: we cannot, by comparison, work out which is more likely to be true. The truth of the gospel-worldview is dependent on the truth of the gospel story. And that cannot be received or believed without God opening blind eyes by the Spirit.