How to properly declare module settings with options list In my ConfigFormBase derived class I want to add select list to settings form. 

How should I declare it in mymod.config.yml file?

I've tried:

mymod.config:
  type: config_object
  label: 'Settings'
  mapping:
    test_mode:
      type: checkbox
      label: 'Turn on test mode'
    country_code:
      type: multiple
      label: 'Action site'
      value:
        - 1
        - 22
        - 34
        - 56
        - 107
        - 168
        - 181
        - 209
        - 228


and can I specify list as key -> value pairs also?
Config schema doesn't know about allowed values. It's a about the structure of the stored configuration, not how it is displayed to the user.

It's up to you to implement the form for that, using e.g. a #type selection and #options.

For the config schema, it's type: boolean, not type checkbox. And type: integer, what you tried to use there doesn't exist.