I am trying to rasterize a vector layer using the "rasterize tool" in QGIS... The problem is that there is no "attribute fields" available to choose from (even though they exist!)...
As you figured out, the column is not defined as 'Integer' or 'Real' type. You can use Field Calculator to add a new column of type integer and write an expression such as below to have a field you can use in conversion.
I had a similar problem although the shapefile was ok. Going to "geoprocessing tools" and using the "gdal" "rasterize" directly instead of using the "rasterize" linked in the Qgis main menu (Raster>Conversions>Rasterize) solved the problem.