Storing a date created with Form API
Drupal 7 Schema API Data types does not support the DATE datatype that you find in popular SQL dialects such as MySQL.  My preferred solution is the same as the one you suggest (set the column to varchar and store it as a an ISO 8601 format date string: 2013-05-07 for May 7, 2013).  It is human readable (useful when debugging) and it is the format used by the DATE format of most SQL dialects.

For date arithmetic, I use built-in PHP-functions such as gregoriantojd to convert to Julian dates to compute date differences, etc.

While it is possible to force Drupal to use the DATE type to your SQL dialect of choice - see this answer for details, a varcharis the simplest and most portable solution.

For displaying a date in Drupal, you may want to use the the Drupal format_date function (from common.inc), as this makes sure that dates are rendered consistent across the site, and adapt month names, etc. to the preferred language.  Note that this function takes a Unix timestamp as its first argument, so you need to convert from the ISO 8601-format to a Unix timestamp in order to use this (and this won't work for dates fore 1970-01-01).  Because of this limitation I instead use a small custom module with a slightly adapted copy of this core function that shares the format templates with the core function, but accepts an ISO 8601 date and datetime as valid input. I also supply my own custom format (you can set up this on the Home » Administration » Configuration » Regional and language  » Date and time  » Add Date type admin page) without any hours, minutes or seconds.

PS: Some people recommend using Unix timestamps (seconds from epoch starting 1970-01-01) for anything that has to to with time and date.  However, I do not like to use Unix timestamps  for dates where the resolution wanted is whole days.  In addition to not being human readable, and having a rather limited timespan (making it inappropriate to store the date of birth for someone born before 1970), Unix timestamp dates based upon the local time tend to produce weird bugs when dates cross seasons because DST is being taken into account when you do not expect it.  It may be considered strange that I depreciate the Unix timestamp when the format_date function wants this format - but I find that storing the dates in the database  in the ISO 8601 format is so attractive that I put up this extra conversion step when calling the format_date function.