REST API presence in Content Server

You can have a look at the /apiinfo resource. I don't know, how far it works for CS core and optional module REST APIs.

There is no resource in the current core CS REST API, which would tell you the CS update version, or the list of installed modules and their version, AFAIK.

How to open a new browser window/tab, when a document is clicked on in the table:

csui.onReady(function () {
  // Specify your server connection here
  var cgiUrl = '...',
      supportPath = '...';
  // Create the widget
  new csui.widget.FolderBrowserWidget({
    connection: {
      url: cgiUrl + '/api/v1',
      supportPath: supportPath
    }
  })
  // Listen to the event, which precedes the default click action execution
  .on('executingDefaultAction', function (args) {
    var node = args.node;
    // Handle documents in a special way
    if (node.type === 144) {
      // Open a new window with the Classic CS UI page
      window.open(cgiUrl + '?func=...&objAction=...&objId=' + node.id);
      // Prevent the default action handling in the widget
      args.cancel = true;
    }
  })
  // Show the widget
  .show({placeholder: '#browser'});
});

<div id="browser"></div>

Well, I'm afraid I wouldn't :-)

There comes no login page with the CS UI Widgets. (Just the fallback dialog you will get, if you don't provide OTCSTicker or OTDSTicket or other authentication header with the server connection parameters.) The idea is, that you log in to the portal, where the page with the Widgets resides - SAP, SharePoint, Oracle, Salesforce etc. You don't want to show the user an additional login. That's why you obtain the ticket for the Widgets (for the CS REST API) on the server side, when you generate the ASP/JSP or other page, where the widget is embedded. Usually, you get an OTDSTicket and render it on the page in a JavaScript variable, which will be used in the Widget initialization.

If your integration page is a static HTML page, not generated by some server-side code, where you can hook in, there is no general OOTB SSO solution by OpenText for the Widgets. You will need some server-side generated page to help you get the ticket. For example, OpenText AGA deploys a request handler to the CS, which helps the integration page to obtain the ticket, if the customer logged in to Office 365. (The trust ACS - ADFS - OTDS - CS)

Unfortunately, the resource /api/v1/ping returns the same response in CS10, CS10.5 and CS16, including all their released updates so far. It cannot be used to detect the version of the CS REST API, or the availability of resources.