i have got 2 Questions.
I'm new to Nintex and using SharePoint 2013.
Unfortunately the Nintex action doesn't allow an option to specify the person checking out the item. Looking further into the Lists.CheckOutFile method at MSDN indicates that it too doesn't provide an option to specify the person to check the item out to.
The only way I can think to possibly accomplish this would be to impersonate the person (perhaps using App Step) at the point of requesting the check-out and then again at the point you wish to check-in the item.
To start a workflow on document by clicking a link is possible in a couple of ways;