According to the wikipedia about Star Trek
Warp drive is a hypothetical faster-than-light (FTL) propulsion system
in the setting of many science fiction works, most notably Star Trek.
A spacecraft equipped with a warp drive may travel at apparent speeds
greater than that of light by many orders of magnitude, while
circumventing the relativistic problem of time dilation. In contrast
to many other fictional FTL technologies, such as a "jump drive" or
the Infinite Improbability Drive, the warp drive does not permit
instantaneous travel between two points; instead, warp drive
technology creates an artificial "bubble" of normal space-time that
surrounds the spacecraft (as opposed to entering a separate realm or
dimension like hyperspace, as is used in the Star Wars, Stargate
franchise, Warhammer 40,000, Babylon 5, Cowboy Bebop and Andromeda
Another explanation of the Warp Drive from the Star Trek Wikia Memory Alpha
Warp drive was a technology that allowed space travel at
faster-than-light speeds. It worked by generating warp fields to form
a subspace bubble that enveloped the starship, distorting the local
spacetime continuum and moving the starship at velocities that could
greatly exceed the speed of light. These velocities were referred to
as warp factors. Warp drive was the most common form of interstellar
propulsion used in the Milky Way Galaxy, making interstellar
exploration, commerce, and warfare possible.
I don't believe there is any canon evidence showing what exact speed they always use when exiting... But from the above mentioned quote I think it's safe to assume that the ship itself can change it's speed during flight because in all technicality they are warping the space around the ship itself. This means that the ship is moving faster than light simply because it's warping the space surrounding the ship. When they exit warp for the most part I have seen they usually use tactics to see what speed they should go when exiting it, such as full impulse if they are trying to get somewhere in a hurry or a slower speed for stealth. That is if they are trying to drop out of warp close to enemies or a dangerous planet or such.
Edit: As for the second question I don't see any reason as to why they can't enter warp at full impulse power because the space around the ship is simply being warped which is what causes the ship to go faster than light.