request.getParameterValues("arrayData")
will return a String Array of all HTTP request parameters named "arrayData". The parameters can be either in the query string or in the HTTP POST body.
However,
$.post('DeleteServlet', {
arrayData : myIdsArray,
mode : "insert"
});
does not create any parameters by this name. That is why you get a null pointer on line 45.
What is sent to the server in this case is a JSON string. What you need to do is parse this string on the server. This will mean (1) Get the string from the POST body. (2) Parse it. You will probably want to use a 3rd-party JSON parser. json-simple is a good one.
Or if you wish to leave your server code as is, you could change the client code to create these request parameters. Probably the easiest approach would be to put them on the query string.