Special Vampire Powers
Do vampires have special powers such as mind reading, ability to compel people or seeing the future?
The simple answer to this is - it depends on the vampire.
Just as humans have certain inherent skills and abilities, so do vampires. However, these inherent skills and abilities are not necessarily the same ones that they had as human beings. The change from human to vampire results in something of a "new birth", and with that comes new abilities and ways of thinking.
Psychic powers such as mind reading and seeing the future are among the more likely of these traits, as are the ability to see spirits and even communicate with the dead. This is most likely because vampires themselves exist somewhere between life and death. Like people who have had near-death experiences (or technically died and were resuscitated), there seems to be a stronger connection to the spirit world with those who have neared it themselves.
Not all vampires have these abilities, though, and most seem to have only one, rather than many. There doesn't seem to be any real logic dictating who gets powers and what those powers might be. Perhaps there is a logic that we just don't understand, but if there is it doesn't seem that vampires understand it much either.
Keep in mind when I am talking about these particular "special powers" I'm not including the primary vampire powers that they all have, such as immortality, super-human speed and strength, extremely rapid healing, the ability to spend long periods of time without oxygen (since they don't really need it), and (as you mentioned) the ability to compel people to do their will.
This last one is commonly thought of as mind-control but it's not really that powerful. Much like a skilled con artist, vampires can subtly manipulate emotions with ease, making it quite easy to get humans to do what they wish. A human isn't "forced" to do what a vampire wants, though, which is part of the reason vampires like to keep their true identity a secret until it suits them. Or until they are hungry.