Why can't Git Bash run my executable?

To run a program in the current directory in bash, you put ./ in front of it. So in your case:

$ ./sqlite3.exe

When you run sqlite3, bash will look for a program with exactly that name in all directories of the PATH environment variable, which by default includes standard locations for executables like /usr/local/bin but not your current directory. See here for more info on that.

It's because you're under a is a runtime environment for gcc, that give you support to binaries native under Windows, but you can run any exe as shell using ./ (local execute) Take a look to documentation of this tool: http://sourceforge.net/p/mingw-w64/wiki2/FAQ/