I finally managed to get it working.
So, first of all, there's definitely something wrong with the webservice. I'm not exactly sure what, and it's not my business to fix it, so I'm not even going to try to pinpoint what it was.
Basically, the getLogbookEvents wasn't receiving parameters because it won't accept any parameters inside the body of the request, period. Looking at the SoapUI raw data being sent I discovered that it was always sending the parameters inside the URL, not the body ...
I have no idea how they got a POST method to behave like a GET method, but as it stands they apparently do. Formatting my nameValuePairs in UTF-8 and adding them to the URL is the (totally backwards and not recommended at all) way I got it to work :)
ArrayList<NameValuePair> nameValuePairs = new ArrayList<NameValuePair>();
nameValuePairs.add(new BasicNameValuePair("startDate", "2014-04-18T05:00:00"));
nameValuePairs.add(new BasicNameValuePair("endDate", "2014-04-18T06:00:00"));
String paramString = URLEncodedUtils.format(nameValuePairs, "utf-8");
HttpPost post = new HttpPost("... basic URL String ..." + paramString);