The first chart shows the region of space from 342 degrees to 354 degrees.
Based on what? The rim has a scale with these running numbers, yes - but the corresponding vertical scale goes from 102 to 106 with a division five times denser. Degrees are a somewhat unlikely unit of measure in either case. And issues of curvature suggest the chart takes significant artistic license, as opposed to really depicting an oddly tilted and compressed pie wedge of a galactic pie, with the galactic core as the vertex and with straight core-centric radial and circular lines denoting the edges of the "sectors". From the star names quoted, we could rather assume that the chart covers only about two degrees of the galactic circle, max.
Still, references to "north" or "east", etc, with regard to directions in the galaxy don't make sense to me. Shouldn't directions be referred to as "spinward" or "antispinward", and towards or away from the galactic core? Then there's out-of-the-plane of the galaxy for the third dimension.
There would need to be an agreed-upon expression for the directions of "up" and "down" from the plane of the galactic disk, yes. And "north" and "south" would serve very logically there, as they would be defined identically to how they are defined on Earth: with respect to the axis/equator system and the direction of rotation. If you look along the axis so that the planet/galaxy spins counterclockwise, you are looking from north.
We already know Star Trek uses "north" and "south" when describing the division of the galaxy: Cheron of "Let That Be, Let That Beeee, Let That Be, Let That Beeee" was indeed said to be "in the southernmost part of the galaxy". This is a practical definition if we assume an Earth-style definition of south, because this would squeeze Cheron into the compact volume of the bottom vertex of the galaxy; if we decided north was towards the core, then the southernmost part would be the vast, indefinite rim of the galactic disk, and not a practical indication of direction at all. (Of course, assuming that Cheron indeed lies directly beneath the galactic core is probably more trouble than worth in the general Trek context...)
The "northeast" reference from TAS makes much less sense. We can define east and west in Earth style, as the respective synonyms for spinward and antispinward. But "northeast" would have to be from a specific spot or "longitude" of the galaxy, then - there would be no rhyme or reason for anything to be "northeast" of the galactic plane.
For the ship to reach the barrier in less than than a year, it would have needed to be traveling at faster than warp factor 30.
We have no idea how far away the Great Barrier would be either from the core or from Earth. For all we know, it blocks travel in the direction of the core within the first few dozen lightyears already. Reaching it would only be the first step on a voyage towards the core - but once you did go there and braved the Barrier, you'd immediately run into the Paradise Prison Planet. (Which in ST5 looked almost exactly like the journey to the core regions in TNG "Nth Degree" did: you hit a scary-looking distortion in space, and suddenly you are at a specific destination far, far away!)
It is possible for the Valiant to circumnavigate the Federation if the ship was traveling at warp factor 9.9 for its entire voyage - could a Defiant-class starship do this?
The
Defiant herself explicitly could not ("The Sound of Her Voice"). She might have been an atypical example of her class, though.
But a journey consisting of short hops between Federation locations of interest could probably be conducted at much higher speeds than a single stretch of journey through wilderness equal to the sum total of the hops. Starships can clearly reach much higher speeds on short hops than over long distances - and even counting in a lot of breathers and pit stops, the end result might favor a piecemeal journey over a sustained one. But only if the pieces were all within civilized space where the pit stops were of practical benefit; stopping in the middle of wilderness would apparently not allow the ship to regain her strength to an equal degree, hence the much lower average speeds quoted for long journeys.
Timo Saloniemi