You need to add the days to the date before you cast it to a string. Try changing
$(function()
{
$('.date-picker').datePicker({clickInput:true}).val(new Date().asString()).addDays(2).trigger('change');
});
to
$(function()
{
$('.date-picker').datePicker({clickInput:true}).val(new Date().addDays(2).asString()).trigger('change');
});
EDIT: Sorry, didn't see the last requirement that today and tomorrow should not be selectable. Try
$(function()
{
$('.date-picker').datePicker({clickInput:true, startDate: new Date().addDays(2).asString()}).val(new Date().addDays(2).asString()).trigger('change');
});