Your question is incredibly general, however I will try to answer it as best as possible.
yes, you most likely can integrate COBOL with your application. It depends on what programming language you want to integrate it with. Just google "Integrate COBOL with (whatever programming language you're using".
Yes, you can create an application in pure COBOL. Many legacy applications use pure COBOL.
I recommend you take a look at this: http://www.netcobol.com/product/netcobol-for-net/
Which can let you compile COBOL on the .net framework.
However, a word of advice: I don't recommend integrating COBOL at all, COBOL is a notoriously bad language to use because it is incredibly verbose. Unless it is for legacy code, I recommend using a more modern language such as a scripting language.