To get the spreadsheet, once you have the DestinationID, use SpreadsheetApp.openById(). Once you have that, you can retrieve an array of sheets, and get the response sheet by index, regardless of its name.
var destId = FormApp.getActiveForm().getDestinationId();
var ss = SpreadsheetApp.openById(destId);
var respSheet = ss.getSheets()[0]; // Forms typically go into sheet 0.
...
From this point, you can manipulate the data in the spreadsheet using other Spreadsheet Service methods.
I also have to determine the number of columns the responses uses. It's not equal to the number of questions in the form. I can count the number of items in the form... (but that doesn't match the spreadsheet)
You're right - the number of current items does not equal the number of columns in the spreadsheet. The number of columns each response takes up in the destination sheet includes any questions that have been deleted from the form, and excludes items that are not questions. Also, the order of the columns in the spreadsheet is the order that questions were created in - as you re-arrange your form or insert new questions, the spreadsheet column order does not reflect the new order.
Assuming that the only columns in the spreadsheet are from forms, here's how you could make use of them:
...
var data = respSheet.getDataRange().getValues(); // 2d array of form responses
var headers = data[0]; // timestamp and all questions
var numColumns = headers.length; // count headers
var numResponses = data.length - 1; // count responses
And your last point is correct, you need to correlate names.
Finally, I think there's no way to relate each question with each column in the sheet, but by comparing somehow columns names with items title.