Context. The Tarantula region in the Large Magellanic Cloud (LMC) contains the richest population of spatially resolved massive O-type stars known so far. This unmatched sample offers an opportunity to test models describing their main-sequence evolution and mass-loss properties. Aims: Using ground-based optical spectroscopy obtained in the framework of the VLT-FLAMES Tarantula Survey (VFTS), we aim to determine stellar, photospheric and wind properties of 72 presumably single O-type giants, bright giants and supergiants and to confront them with predictions of stellar evolution and of line-driven mass-loss theories. Methods: We apply an automated method for quantitative spectroscopic analysis of O stars combining the non-LTE stellar atmosphere model fastwind with the genetic fitting algorithm pikaia to determine the following stellar properties: effective temperature, surface gravity, mass-loss rate, helium abundance, and projected rotational velocity. The latter has been constrained without taking into account the contribution from macro-turbulent motions to the line broadening. Results: We present empirical effective temperature versus spectral subtype calibrations at LMC-metallicity for giants and supergiants. The calibration for giants shows a +1kK offset compared to similar Galactic calibrations; a shift of the same magnitude has been reported for dwarfs. The supergiant calibrations, though only based on a handful of stars, do not seem to indicate such an offset. The presence of a strong upturn at spectral type O3 and earlier can also not be confirmed by our data. In the spectroscopic and classical Hertzsprung-Russell diagrams, our sample O stars are found to occupy the region predicted to be the core hydrogen-burning phase by state-of-the-art models. For stars initially more massive than approximately 60 M⊙, the giant phase already appears relatively early on in the evolution; the supergiant phase develops later. Bright giants, however, are not systematically positioned between giants and supergiants at Minit ≳ 25 M⊙. At masses below 60 M⊙, the dwarf phase clearly precedes the giant and supergiant phases; however this behavior seems to break down at Minit ≲ 18 M⊙. Here, stars classified as late O III and II stars occupy the region where O9.5-9.7 V stars are expected, but where few such late O V stars are actually seen. Though we can not exclude that these stars represent a physically distinct group, this behavior may reflect an intricacy in the luminosity classification at late O spectral subtype. Indeed, on the basis of a secondary classification criterion, the relative strength of Si iv to He i absorption lines, these stars would have been assigned a luminosity class IV or V. Except for five stars, the helium abundance of our sample stars is in agreement with the initial LMC composition. This outcome is independent of their projected spin rates. The aforementioned five stars present moderate projected rotational velocities (i.e., νesini < 200kms-1) and hence do not agree with current predictions of rotational mixing in main-sequence stars. They may potentially reveal other physics not included in the models such as binary-interaction effects. Adopting theoretical results for the wind velocity law, we find modified wind momenta for LMC stars that are 0.3 dex higher than earlier results. For stars brighter than 105 L⊙, that is, in the regime of strong stellar winds, the measured (unclumped) mass-loss rates could be considered to be in agreement with line-driven wind predictions if the clump volume filling factors were fV 1/8 to 1/6. Based on observations collected at the European Southern Observatory under program ID 182.D-0222.Tables C.1-C.5 are also available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/600/A81