Is the “chsh” command available in Cygwin

Use c:\cygwin\bin\run.exe to launch cygwin processes from windows cmd.exe

The terminal can't find my .minttyrc config file

You can use the -c file option to specify the location of the mintty configuration file.

The easiest way to do this is to change your mintty shortcut to

C:\cygwin\bin\mintty.exe -c ~/.minttyrc -i /Cygwin-Terminal.ico -

Modify as appropriate for your Cygwin setup.

Further reading

$ mintty --help
Usage: mintty [OPTION]... [ PROGRAM [ARG]... | - ]

Start a new terminal session running the specified program or the user's shell.
If a dash is given instead of a program, invoke the shell as a login shell.

Options:
  -c, --config FILE     Load specified config file
  -e, --exec            Treat remaining arguments as the command to execute
  -h, --hold never|start|error|always  Keep window open after command finishes
  -i, --icon FILE[,IX]  Load window icon from file, optionally with index
  -l, --log FILE|-      Log output to file or stdout
  -o, --option OPT=VAL  Override config file option with given value
  -p, --position X,Y    Open window at specified coordinates
  -s, --size COLS,ROWS  Set screen size in characters
  -t, --title TITLE     Set window title (default: the invoked command)
  -u, --utmp            Create a utmp entry
  -w, --window normal|min|max|full|hide  Set initial window state
      --class CLASS     Set window class name (default: mintty)
  -H, --help            Display help and exit
  -V, --version         Print version information and exit