How do the Planets and Sun get their initial rotation?
Why do Venus and Mercury rotate so slowly compared to other planets?
Why does Venus rotate in a different direction to Mercury, Earth and Mars?
Anglular momentum is conserved, so any tiny initial rotation that a the original ball of gas had becomes faster as the gas collapses down into a star and disk of planets.
Planets near the sun rotate slowly for the same reason that the moon always faces the same side to the Earth - tidal braking
Venus probably received a hit from a some lump of rock / proto-planet some time early in it's life which changed it's rotation. A similar event split the moon off from Earth
To build on Martin Beckett's answer (especially because I am not sure how familiar you are with physics);
Currently we believe Stars form when objects known as Molecular Clouds (which are as one might guess, clouds of molecules in space, mostly comprised of hydrogen) collapse. It is important to note that these clouds are not 'static', they have some kind of motion, including some kind of 'average rotation', which is to say that overall the cloud is rotating (usually fairly slowly).
As was mentioned in Martin Beckett's answer, angular momentum is conserved; the typical example to give is to imagine a spinning figure skater, as she brings her arms in close to her body, she spins faster. If you don't believe this and have access to an office chair, it is easy to convince yourself (and possibly injure yourself too...). This holds true for the molecular cloud as well. As it collapses in on itself, it starts rotating faster and faster, forming a disc. The bulk of this coalesces into a big ball of hydrogen at the centre, which will eventually form a star. The matter in the disc slowly starts to clump together more and more to form the planets (it's a little more complicated than this, but if you're interested it's an easy topic to read up on). Similarly to how the overall cloud starts spinning faster and faster, the matter that forms these planets was spinning and maintains its spin as it clumps together into planets.
The previous post has covered your other questions.
If we leave out Earth for a moment, it appears that rocky planets slow down linearly on their rotation periods - the closer they become to the Sun. Earth is an anomaly probably because of the Moon which has speeded up its rotation up considerably. According to my graph of this linear relationship between 'day' length and distance from the Sun, an Earth without the Moon would have days which are 1,960 hours long. So the Sun must have some influence on the rotation periods of each planet.
A simple answer is conservation of angular momentum. The correct answer is that simple answer is a bit too simple.
There's a problem with this nice simple explanation: While the Sun accounts for 99.87% of the solar system's total mass, the Sun accounts for very little of the solar system's total angular momentum. This "angular momentum problem" (e.g., see Richard Larson, "Angular momentum and the formation of stars and black holes," Reports on Progress in Physics 73.1 (2010): 014901) is even larger during the formation process. The total angular momentum of a typical gas cloud is multiple orders of magnitude larger than is the largest possible angular momentum that a star could sustain, lest it break up. A protostar needs to shed angular momentum as it grows. It doesn't quite make sense to invoke conservation of angular momentum on a system that does not conserve angular momentum. Various proposed processes for this shedding do however more or less keep the axis of rotation pointing in more or less the same direction.
The terrestrial planets
If this hypothesis is correct, the Earth's would have rotated significantly faster than the current rotation rate of one rotation every 24 hours, instead rotating once every four to six hours. Our current rotation rate is a result of a gradual transfer of angular momentum from the Earth's rotation to the Moon's orbit. There are signs of this transfer in fossils and in rocks that show that the Earth did indeed rotate faster than it does now.
Mars however is rather close to Jupiter, and if the Grand Tack model is correct, it was even closer to Jupiter in the distant past. There is strong evidence that Mars' obliquity (axial tilt) has undergone chaotic variations in the past. Mars is yet another special case.
The giant planets