In my opinion it is possible but it can not be as accurate as the analyzing with longer data bases. In general, it is highly recommended to use at least 30 years for climate projections. For example in this article "Climate change outlook for water resources management in a semiarid river basin: the effect of the environmental water demand", I used 30 years for observation data (1980-2009).
I agree that it is not a good idea. To have a robust training period, 30 years of continuous data from one source is ideal. Combining such different datasets should not be done either. Gridded products generally do not have the same variability as observed station data, and depending on the location, could be quite different. This is especially true for extreme events, which can vary widely on very small scales.
Thank you all ( Anne Marie Koch Stoner , Marcelo Eduardo Rojas Vidal , Parisa Sarzaeim ) for your valuable insights. It seems I am stuck. In addition, if I may ask:
can I use CORDEX RCM without biased correction for a medium sized watershed (900 km^2) in an Eastern Himalayan region?
Ngahorza, for a watershed that size I would say yes, you can use CORDEX, especially since that is pretty much all you have and bias correcting it is not an option. Just be sure to make a note of it in any publication etc. Some extremes may be somewhat muted as they are with all gridded datasets.
There is also the NASA NEX-GDDP downscaled output you might consider (https://cds.nccs.nasa.gov/nex-gddp/). It's global at 0.25 degree resolution.