Where can I download GIS data delimiting global climate types? For example mediterannean climate, temperate climate, etc.
Additionally, how similar are temperature profiles for various places in the same climate classification?
NOAA has climate division files in netcdf and shapefile format, though they may be just CONUS:
The most promising link looks like this source with paper citation below. They have lots of freely available data in a variety of formats:
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:
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: