I've installed Piwik 2.0-b11 and have performed the same test. These problems still exist, in fact I am noticing additional strange behavior.
Referencing the results for Germany in the following reports:
2013-11-01 to 2013-11-30:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-11-30&format=xml&token_auth=anonymous
nb_visits = 5380, sum_daily_nb_uniq_visitors = 4759
Add one day 2013-11-01 to 2013-12-01:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-12-01&format=xml&token_auth=anonymous
Empty result set.
Add another day 2013-11-01 to 2013-12-02:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-12-02&format=xml&token_auth=anonymous
nb_visits = 5696, sum_daily_nb_uniq_visitors = 289 (!?)
This appears to be a problem more global in nature, as it is not limitted to only the UserCountry methods. I found the same problem exists in DevicesDetection, UserSettings, and Provider methods as well.
Referencing the results for Germany in the following reports:
2013-11-01 to 2013-11-30:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-11-30&format=xml&token_auth=anonymous
nb_visits = 5380, sum_daily_nb_uniq_visitors = 4759
Add one day 2013-11-01 to 2013-12-01:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-12-01&format=xml&token_auth=anonymous
Empty result set.
Add another day 2013-11-01 to 2013-12-02:
http://demo.piwik.org/?module=API&method=UserCountry.getCountry&idSite=7&period=range&date=2013-11-01,2013-12-02&format=xml&token_auth=anonymous
nb_visits = 5696, sum_daily_nb_uniq_visitors = 289 (!?)
This appears to be a problem more global in nature, as it is not limitted to only the UserCountry methods. I found the same problem exists in DevicesDetection, UserSettings, and Provider methods as well.