The Arduino IDE sometimes has difficulties with this, so I would not keep my hopes up. Depending on the board this changes every time the board is reset or flashed. Because COM ports are not used much anymore nowadays, I would just ask Windows to give me all available and if there is more than one, ask the user to select the right one.
Regarding the Windows code, looks like when you Google "C# com port" you get some nice examples from Microsoft in the first hit. 