GIS Data on climate types
NOAA has climate division files in netcdf and shapefile format, though they may be just CONUS:

http://www.esrl.noaa.gov/psd/data/usclimdivs/boundaries.html

The most promising link looks like this source with paper citation below.  They have lots of freely available data in a variety of formats:

http://koeppen-geiger.vu-wien.ac.at/shifts.htm

Rubel, F., and M. Kottek, 2010: Observed and projected climate shifts 1901-2100 depicted by world maps of the Köppen-Geiger climate classification. Meteorol. Z., 19, 135-141. DOI: 10.1127/0941-2948/2010/0430.

There's also the ArcGIS Online map service that has Koppen-Geiger Observed and Predicted Climate Shifts:

http://www.arcgis.com/home/item.html?id=0b9f1f90f4f745daacbc35599609b59a

To answer your question about the types of temperature in a classification,  I suggest you refer to the paper above and the ranges of expected temperatures for each Koppen climate type which can be found here: https://en.wikipedia.org/wiki/K%C3%B6ppen_climate_classification 

For modeled and observed temperature, precipitation, etc. you might find the following resources helpful:

http://iridl.ldeo.columbia.edu/

http://gisclimatechange.ucar.edu/

http://www.worldclim.org/