Unless he started in North Africa (pretty sure he started on the island) then there is definitely some significant space displacement, probably about 10,000 miles as the crow flies. Unless he started in late 2005 (he arrives in Tunisia October 24, 2005) then there was significant time displacement, about 10 months.

So to answer your first question of whether his "jump is primarily through time or space one needs to discern which is greater 10,000 miles or 10 months?

I don't think the "jump" is perfected in this case, but obviously in others (e.g. Tom to NYC, Richard and Ethan to Florida and Portland) it was. Perhaps Ben was in a hurry or something went wrong. Note the injury in your last question/observation.

I don't think he had specific landing expectations, more of an escape. He's such an opportunist/survivalist (as time travelers must be) he quickly adapted.

Yes the injury is important. I originally thought that when he went into the secret room to summon Smokey he also made this trip to Tunisia and then to London all as a reaction to Alex's death. But maybe that trip was later he was all cleaned up and uninjured in London? Also if he went at the time he went into the room did the injury occur during the journey? Very tricky, something similar may be going on with Dr. Ray's injury and various time frames.

Good questions wish I had answers.