If any one can read Icelandic (which I can't I'm afraid!), there is a website here which gives some info and pictures and diagrams on the volcano and the area. There is some info on the volcano and the area in English here on the tour website. The video below gives a good overview of the size of the volcano and the gap underneath.
I can't comment with any certainty on the matter having not visited myself (its always best to look at things in the flesh and be there, you get a much clearer picture and can make clearer judgements for yourself!), however I am going to put my 2 cents in about the crater itself anyway.
As a cinder cone the type of activity at Thrihnukagigur volcano will be much like some of the episodes at Etna which have created cinder cones there over the years, and as such, present an interesting incite into processes below the surface during such events. The vivid colours inside the volcano demonstrate the high gas content and activity of the magma, further confirming the type of activity there. In certain volcanoes worldwide there are lava lakes and these can be seen to ebb and flow in height and activity, therefore we can accept that other volcanoes in similar situations will do this too and can often be seen as 'tide marks' of sort. It is certainly fortunate for us that the magma drained away (which there could be a few reasons why this happened) as we now have access to something quite beautiful and it does provide scope for research into the shallow plumbing systems of volcanoes.