Solution 1:

The Simple Solution

There is no need to convert Strings to Dates or use RegExp.

The simple solution is to use the Array.sort() method. The sort function sets the date format to YYYYMMDD and then compares the string value. Assumes date input is in format DD/MM/YYYY.

data.sort(function(a,b) {
  a = a.split('/').reverse().join('');
  b = b.split('/').reverse().join('');
  return a > b ? 1 : a < b ? -1 : 0;
  // return a.localeCompare(b);         // <-- alternative 
});

Update:

A helpful comment suggested using localeCompare() to simplify the sort function. This alternative is shown in the above code snippet.

Run Snippet to Test

<!doctype html>
<html>
<body style="font-family: monospace">
<ol id="stdout"></ol>
<script>
  var data = ["09/06/2015", "25/06/2015", "22/06/2015", "25/07/2015", "18/05/2015"];

data.sort(function(a,b) {
  a = a.split('/').reverse().join('');
  b = b.split('/').reverse().join('');
  return a > b ? 1 : a < b ? -1 : 0;
  
  // return a.localeCompare(b);         // <-- alternative 
  
});

for(var i=0; i<data.length; i++) 
  stdout.innerHTML += '<li>' + data[i];
</script>
</body>
</html>

Solution 2:

You will need to convert your strings to dates, and compare those dates, if you want to sort them. You can make use of the parameter that the sort method accepts, in order to achieve this:

var dateStrings = ["09/06/2015", "25/06/2015", "22/06/2015", "25/07/2015", "18/05/2015"];
var sortedStrings = dateStrings.sort(function(a,b) {
    var aComps = a.split("/");
    var bComps = b.split("/");
    var aDate = new Date(aComps[2], aComps[1], aComps[0]);
    var bDate = new Date(bComps[2], bComps[1], bComps[0]);
    return aDate.getTime() - bDate.getTime();
});

In order to reduce code redundancy, and to handle different date formats, you can add an additional function that will create the comparator needed by the sort method:

function createSorter(dateParser) {
    return function(a, b) {
        var aDate = dateParser(a);
        var bDate = dateParser(b);
        return aDate.getTime() - bDate.getTime();
    };
}

dateStrings.sort(createSorter(function(dateString) {
    var comps = dateString.split("/");
    return new Date(comps[2], comps[1], comps[0]);
}));

You can then use different date formatters by passing different functions to the createSorter call.

As for your second question, you can create an (sorted) array of dates from your strings, and perform your logic on that array:

function myDateParser(dateString) {
    var comps = dateString.split("/");
    return new Date(comps[2], comps[1], comps[0]);
}

var sortedDates = dateStrings.map(myDateParser).sort();

You can walk through the sortedDates array and if you find two non-consecutive dates, then you have dates with gaps between them.