Looks like you're off and running. A couple of loose ends. It looks like George enlisted too late to have been serving in the 1st NC Infantry (which disbanded in Nov 1861)..... I removed the top sticker Added a few others...... I'm done adding so she's all yours.
Regarding Company A vs Company E..... The National Park's service site (and the Family search source that's posted) say Company E....... National Parks Service “The Civil War” https://www.nps.gov/civilwar/search-soldiers-detail.htm?soldierId=7FA330BE-DC7A-DF11-BF36-B8AC6F5D926A Provides: Pvt George A Neal served in Company E 11th Regiment, North Carolina Infantry (Bethel Regiment)
I also found a secondary site that says Company A...... Civil War Records of North Carolina by CensusDiggins.com ; http://www.censusdiggins.com/nc_civil_war_n.html Provides: G.A. Neal Co A, 11th NC Infantry
Enlisted 2-1-62 at Mecklenburg Co, NC. Private, age 18. Wounded & POW in July 1863 at Gettysburg, PA. In hospital 9-29-64 at Raleigh, NC and had his left arm amputated. 60 day furlough 10-7-64.
You get to decide (might have to solve this by actually seeing Muster Rolls...) but keep in mind, he won't be with his buddies if you get the wrong company.