I'm trying to do spatial analysis of the Caspian Sea area and am not sure which projection to use for the countries around it. The Caspian spans multiple UTM zones and the countries around is are from two continents so I'm hesitant to use one Continental projection. Is it small enough scale picking one of the UTM zones wouldn't distort the countries too much? Just looking for the best projection to use and why.
I generally use projection wizard when I want to begin thinking about which projection to choose:
You can choose your extent and it will give you recommendations as to the best projection to use. I don't know how big your extent is (Russia borders the Caspian sea, are you mapping all of it?), so I can't recommend a specific projection. I tend to stay away from UTM projections if I am traversing more than two zones.
As to why you would want to use one projection over another, you should think about both the area you are attempting to analyze (projection wizard can help with this) and the end product of your analysis. Generally, projections can be split into:
1) Conformal, which preserves angles and is typically used for navigation and display of most maps;
2) Equidistant, which preserves relative distances, and can be used for measuring length (though not all lengths will be preserved); and
3) Equivalent, which preserves area and is used for the measurement of areas.
For most of my spatial analyses, I use equivalent projections. For most cartographic projects, I use conformal projections. For your project, do you care most about relative positioning (conformal), relative distances (equidistant), or areas (equivalent)? The answer to this will help you select between your options.
The only projected CRS about the Caspian Sea available in the EPSG Registry v. 9.2 is "Pulkovo 1942 / Caspian Sea Mercator" (EPSG:3388) and it's suitable to do spatial analysis.