Do black holes accelerate in spin as they obtain more mass? It is known that - When a star collapses during the formation of the black hole, the black hole obtains the spin of the star which it collapsed from...
What I'd like to know is, If this spin accelerates as a result of angular momentum (if any), What effects could this rapid rotation have on the black hole, its gravity or anything else around it?
 A: The black hole gains the angular momentum of the matter that falls into it, minus the anglar momentum of the outgoing gravitational radiation.  There are a lot of questions currently on the site regarding the effects of a spinning black hole.  The primary one is that the black hole will cause the space around it to co-rotate with the hole.
A: This is question is both open theoretically and experimentally, although there is some astrophysical data.
After a black hole forms, the matter is squeezed as it falls into an accretion disk, and the matter that is absorbed is rotating very fast by the time it falls in. The dynamics of the accretion disk are complicated, because there are horrendous fields caused by ionization and charge separation. This makes it difficult to achieve consensus about whether the matter falling into a small black hole causes its spin to slow down or speeds up.
The experimental consensus seems to currently be that the black holes that power active galactic nuclei are spinning at close to extremality, meaning they have close to the maximum amount of angular momentum for their mass. One reason to expect this is that, absent quantum mechanics, the black hole can only generate energy to the extent that it is spinning. The AGN's are generating a huge amount of energy, so something must be maintaining the spin, and this should be the infalling matter.
It is my opinion that there is still uncertainty regarding the emission of highly spinning black holes, because there is no certainty about what happens to infalling matter. One has reasons to suspect that this matter is emitted from the black hole nonthermally, more or less as it came in, after doing a traversal of the interior regions. If this is so, you must take into account the spin-up/spin-down effects of the in-out matter, and this requires the classical limit of quantum black hole, something which is nearly, but not quite, available, thanks to string theory.
