I'm a bit confused by some of this.
First up:
$countries[]=<?php foreach($country as $value)echo "$value."; ?><br />
Isn't valid code:
To define an array you would write
$countries = Array();
However, PHP is nice and if you set $countries = to an array value it automatically creates an array.
But, you've also declared $countries[] outside of the PHP block so that won't work. All your PHP work has to between <?php and ?>.
Yes you're correct in the sense that you seem to be mixing countries and country variables.
I'm not sure about how your multiple selector will work (as I say, I use checkboxes) but theoretically after you've submitted your form you'll need to do something like this:
<?php
$countries = $_POST['country']; // To retrieve the multi select values
// Then open your db connection.
// Then for loop over $countries
// For each value in $countries you'll want to do a separate insert statement.
// Obviously if you loop using variable $i you'll need to specify $countries[$i] in your INSERT command, not $countries on its own.
?>
Oh and note that MySQL expects the date in the form YYYY-MM-DD so you'll need to convert. You can do this with date("Y-m-D", mktime()), I think. Check the documentation.