Are Rorschach (inkblot) tests still used as a diagnostic tool?
Yes, they are still in use, and they provide rather bad information. My former advisor once worked with a clinician who told him that the main reason they're any better than using the weather report as a projective stimulus is that the Rorschach tests have been in use as-is for decades. Hence we have a better understanding of how people normally react to them, and some catalog of unusual reactions that have been associated with particular characteristics. Had we started with a set of weather reports and stuck with them though, maybe not much else would be different.

Anyway, here are some excerpts from Wikipedia about usage:


  United States
  
  The Rorschach test is used almost exclusively by psychologists. In a survey done in the year 2000, 20% of correctional psychologists used the Rorschach while 80% used the MMPI.[8] Forensic psychologists use the Rorschach 36% of the time.[79]
  In custody cases, 23% of psychologists use the Rorschach to examine a child.[80]
  Another survey found that 124 out of 161 (77%) of clinical psychologists engaging in assessment services utilize the Rorschach,[81] and 80% of psychology graduate programs teach its use.[9]
  Another study found that its use by clinical psychologists was only 43%, while it was used less than 24% of the time by school psychologists.[79]
  
  United Kingdom
  
  Many psychologists in the United Kingdom do not trust its efficacy and it is rarely used.[82] Although skeptical about its scientific validity, some psychologists use it in therapy and coaching "as a way of encouraging self-reflection and starting a conversation about the person's internal world."[25] It is still used, however, by such prestigious mental health organisations as the Tavistock Clinic.[83]
  
  Japan
  
  Shortly after publication of Rorschach's book, a copy found its way to Japan where it was discovered by one of the country's leading psychiatrists in a second-hand book store. He was so impressed that he started a craze for the test that has never diminished. The Japanese Rorschach Society is by far the largest in the world and the test is "routinely put to a wide range of purposes".[26] The test has recently been described as "more popular than ever" in Japan.[83]


And about the objects of (intended / purported) measurement:


  The general goal of the test is to provide data about cognition and personality variables such as motivations, response tendencies, cognitive operations, affectivity, and personal/interpersonal perceptions. The underlying assumption is that an individual will class external stimuli based on person-specific perceptual sets, and including needs, base motives, conflicts, and that this clustering process is representative of the process used in real-life situations.[33] Methods of interpretation differ. Rorschach scoring systems have been described as a system of pegs on which to hang one's knowledge of personality.[34] The most widely used method in the United States is based on the work of Exner.




References
8. Raynor, P., &amp; McIvor, G. (2008). Developments in social work offenders (Research highlights in social work) (pp. 138). London: Jessica Kingsley Publishers. ISBN: 1-84310-538-1.
9. Weiner, I. B., &amp; Greene, R. L. (2007). Handbook of personality assessment. John Wiley &amp; Sons. ISBN: 0-471-22881-8.
25. BBC News. (2012, July 24). What's behind the Rorschach inkblot test? BBC News Magazine. Retrieved July 31, 2014 from http://www.bbc.com/news/magazine-18952667.
26. Fidgen, J. (2012, July 25). On A. Hall (Producer), Dr Inkblot. BBC Radio 4. Retrieved July 31, 2014 from http://www.bbc.co.uk/programmes/b01l0kch.
33. Groth-Marnat, G. (2003). Handbook of psychological assessment. John Wiley &amp; Sons. ISBN: 978-0-471-41979-2.
34. Mons, W. (1950). Principles and practice of the Rorschach Personality Test (2nd ed., pp. 30–31). Faber.
79. Hughes, Gacono, C. B., &amp; Owen, P. F. (2007). Current status of Rorschach assessment: Implications for the school psychologist. Psychology in the Schools, 44(3), 281. DOI: 10.1002/pits.20223.
80. Butcher, J. N. (2009). Oxford handbook of personality assessment (Oxford Library of Psychology) (pp. 290).  Oxford University Press. ISBN: 0-19-536687-5.
81. Camara, Nathan, J. S., &amp; Puente, A. E. (2000). Psychological test usage: Implications in professional psychology. Professional Psychology: Research and Practice, 31(2), 131–154. DOI: 10.1037/0735-7028.31.2.141.
82. Guardian staff. (2013, November 8). Hermann Rorschach Google doodle asks users to interpret inkblot test. The Guardian. Retrieved July 31, 2014 from http://www.theguardian.com/technology/2013/nov/08/hermann-rorschach-google-doodle-inkblot-test.
83. Hall, A. (2012, July 25). Dr Inkblot. BBC Radio 4. Retrieved July 31, 2014 from http://www.bbc.co.uk/programmes/b01l0kch.