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I don't believe they are the same Carl Kumm. My Carl immigrated in 1874, after your Carl fought in the Civil War. My Carl is buried in the Hillside Cemetery in Columbus, WI with his wife, Wilhemina. For further confirmation, you could check the birthdates of the 13 children against the birthdates of my Carl's children - Nov 6, 1858/7, Aug 8, 1860, Jan 21, 1864, and Jul 6, 1866. I know for sure Emilie was born in 1858 in Mittenwalde, Prussia, Germany.
I would not rule out that our two Carls are related, your Carl in Wisconsin could have gotten word back to my Carl in Germany and persuaded him to immigrate. Although I believe Mittenburg is a long way from Baden Baden.