The Messenger craft will have to do 3 flybys of Mercury before it can go into orbit. Is the reason due to Mercury's high orbital velocity and weak gravity?

Any craft leaving Earth and moving into the inner solar system would have to have a slower orbital speed than the Earth so that it can "fall" into the Sun's gravity well.

To achieve this is the craft launched in a direction opposite to Earth's movement?

Please discuss the orbital dynamics of entering orbit around Mercury, a planet with a high orbital speed and weak gravity.

Does entering Venus orbit present similar challenges?

Here is a link to lots of information about the mission.
http://messenger.jhuapl.edu/the_miss...on_design.html
For the last question, yes. That slows the craft down so that it falls into an orbit that takes it to Venus and then Mercury.
On the first flyby the craft is near perihelion of a long elliptical orbit, and is moving much faster than Mercury. The trajectory is chosen so that the planet's gravity gives the craft a retrograde slingshot effect that makes the next flyby slower. A similar sequence makes the third flyby slower yet. Finally, the fourth approach is slow enough that the craft has enough fuel to make the final retrograde burn and achieve orbit around the planet. The same relatively weak gravity that made three flybys necessary to slow down the approach makes the insertion relatively easy.
For Venus, the approach would be slower, but the insertion burn would need to be stronger because of the stronger gravity. It looks like a tradeoff. I have no means of calculating the relative amounts.

And because it had to shed all the orbital velocity it got from Earth's orbit.

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Just thinking out loud:

First of all you have to achieve Earth escape velocity~11kms

Then you have to enter an elliptical orbit with perihelion at Mercury and aphelion at Earth - approximate average orbital velocity I guess is very roughly average of both planet's orbital velocity (30kms Earth, 48kms Mercury) so around 39 kms.

I guess it is the total delta V budget of over 20 kms that is the killer.

Hmm, orbital velocity is fastest at perihelion and slowest at aphelion so that might help. Possibly not that big a burn to enter the orbit from Earth. I guess the dynamics of this orbit are quite important for working out delta V requirements.

Then when you get to Mercury you have to drop below Mercury escape velocity of 4.25 kms.

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