star trek - Does warp travel happen in the 4th dimension of euclidean space?

No.

Within the Trek universe, ships travelling at warp speed are still capable of interacting with normal three-dimensional matter (with potentially disastrous consequences). This is why ships use a series of nested static shields known as navigational deflectors to move particulate matter out of the way, tractor beams to move medium-sized objects and sensors to detect and avoid larger objects that could prove a threat.

In TNG: Best of Both Worlds, Part II, Riker orders a collision course at warp, implying that this would have a more devastating effect than a collision at sublight speeds:

RIKER : Mister Crusher, ready a collision course with the Borg ship

Wesley reacts, turns and looks for confirmation...Repeating --

RIKER You heard me. A collision course.

WESLEY : Yessir.

RIKER : Mister La Forge, prepare to go to warp power...

and in Star Trek : The Motion Picture the crew have to take evasive action to avoid hitting an asteroid while travelling at warp:

DECKER : Negative control from inertial lag will continue 22 point five seconds before forward velocity slows to sub-light speed.

ILIA
Unidentified small object has
been pulled into the wormhole
with us, Captain! Directly
ahead...!

KIRK : Forcefields up full! Put object on viewer...!

The picture is switched through two further levels of
magnification, enlarging the object: an elongated,
distorted, pitted asteroid, tumbling toward the Enter-
prise on a collision course.

CHEKOV
(punching button)
Torpedoes away...!

EXT. SPACE - PAST THE ENTERPRISE AND THE ASTEROID

    as the starship's photon torpedo tubes EJECT GLOWING
    BALLS OF LIGHT ENERGY, which seems to float toward the
    oncoming asteroid, almost too slowly. And in these
    brief seconds, the asteroid hurtles at the Enterprise,
    the huge pitted rock growing even larger than the ship
    itself. It FILLS THE SCREEN, as the photon torpedoes
    hit, disintegrating the asteroid into thousands of
    fragments. Instantly, these fragments pulverize them-
    selves on the ship's forward forcefield and deflector
    screens. The smaller pieces burn up on impact, clearly
    outlining the ship's forcefield barriers.

INT. BRIDGE - INCLUDING MAIN VIEWER

    The asteroid fragments still smashing into the force
    field screen; the smaller bits like SPARKLERS as im-
    pact heat consumes them. The larger sections bouncing 
    away, the bridge QUIVERING as they hit.

    Then one final gigantic fragment strikes, the bridge
    SHUDDERS. And then the viewer shows only the normal
    SUB-WARP EFFECT: The stars ahead, relatively station-
    ary; a feeling of motion, but smooth, visually normal.

As regards the usefulness of the Navigational Deflector, n at least one episode of Voyager (Year of Hell, Part II) do we see the effects of it becoming inoperative (albeit in this case it's the objects that are moving at speed, not the ship)

KIM: Captain, with the deflector down those micrometeoroids are beginning to erode the hull.

JANEWAY: Emergency power to the deflector.

TUVOK: None available.