Hi Davyd, I have bad news for you.
As I said, not all of Drupal modules will work, and Date is one of them.

I setup an environment like yours and I see that Date module is not SQLite aware at all.
A quich search for 'mysql' string returned a bunch of results from the Date module.
File date/date_api_sql.inc contains a lot of switch constructs due to the fact it is strictly DB-related.

Example, at line 10:

function date_sql_concat($array) {
  global $db_type;
  switch ($db_type) {
    case('mysql'):
    case('mysqli'):
      return "CONCAT(". implode(",", $array) .")";
    case('pgsql'):
      return implode(" || ", $array);
  }
}

This code is easily patchable to make it work with SQLite:

function date_sql_concat($array) {
  global $db_type;
  switch ($db_type) {
    case('mysql'):
    case('mysqli'):
      return "CONCAT(". implode(",", $array) .")";
    case('sqlite'):
    case('pgsql'):
      return implode(" || ", $array);
  }
}

This is the easiest one, other functions are harder.
See sql_field() at line 163, which requires deep changes and, maybe, missing SQLite SQL functions to be added.

I think that making Date module work with Drupal-SQLite is not feasible.
Sorry.