In this paper we present the first results of a study of BAL QSOs (at low and high redshift), based on very deep Gemini GMOS integral field spectroscopy. In particular, the results obtained for the nearest BAL IR-QSO Mrk 231 are presented. For the nuclear region of Mrk 231, the QSO and host galaxy components were modelled, using a new technique of decoupling 3D spectra. From this study, the following main results were found: (i) in the pure host galaxy spectrum an extreme nuclear starburst component was clearly observed, as a very strong increase in the flux, at the blue wavelengths; (ii) the BAL system I is observed in the spectrum of the host galaxy; (iii) in the clean/pure QSO emission spectrum, only broad lines were detected. 3D GMOS individual spectra (specially in the near-infrared CaII triplet) and maps confirm the presence of an extreme and young nuclear starburst (8 < age < 15 Myr), which was detected in a ring or toroid with a radius r = 0.3arcsec ~ 200 pc, around the core of the nucleus. The extreme continuum blue component was detected only to the south of the core of the nucleus. This area is coincident with the region where we previously suggested that the galactic wind is cleaning the nuclear dust. Very deep 3D spectra and maps clearly show that the BAL systems I and II - in the strong `absorption lines' NaIDλ5889-95 and CaII Kλ3933 - are extended (reaching ~1.4-1.6 arcsec ~ 1.2-1.3 kpc, from the nucleus) and clearly elongated at the position angle (PA) close to the radio jet PA, which suggest that the BAL systems I and II are `both' associated with the radio jet. The physical properties of the four expanding nuclear bubbles were analysed, using the GMOS 3D spectra and maps. In particular, we found strong multiple LINER/OF emission-line systems and Wolf-Rayet features in the main knots of the more external super bubble S1 (r = 3.0 kpc). The kinematics of these knots - and the internal bubbles - suggest that they are associated with an area of rupture of the shell S1 (at the south-west). In addition, in the more internal superbubble S4 and close to the core of the nucleus (for r < 0.7arcsec ~ 0.6 kpc), two similar narrow emission-line systems were detected, with strong [SII] and [OI] emission and ΔV ~ -200 km s-1. These results suggest that an important part of the nuclear NLR is generated by the OF process and the associated low-velocity ionizing shocks. Finally, the nature of the composite BAL systems and very extended OF process - of 50 kpc - in Mrk 231 (and similar QSOs) are discussed. In addition, the `composite hyperwind scenario' (already proposed for BALs) is suggested for the origin of giant Lyα blobs. The importance of study the end phases of Mrk 231, and similar evolving elliptical galaxies and QSOs (i.e. galaxy remnants) is discussed.