I can only definitively answer part (1), and that answer is:
Yes, once you have explored a system you retain that information and it shows on the System Map for you.
I understand from the game forums that the first players to survey a system get more income from doing so than later players - but that later players do get some (presumably this data is confirmation of the earlier surveyors' findings).
I recall from memories of statements by the developers (can't cite, sorry) that in the released game, systems that have been explored and see a lot of player traffic will eventually have space stations built and come under faction control. I don't know whether the faction that players sell the data to has influence on the faction that moves into the system.
The requirements for successful docking are:
- Docking permission granted, and not expired.
- Landing gear deployed.
- Being on the correct pad (this can be tough at some outpost stations).
- Being centered on the pad (so that it lights up blue on your sensor display). If you're lined up on one axis and not the other, the lined up axis will light up blue.
- Being reasonably level, with the bottom of your ship on the pad.
- Facing "forward" on the pad (so that in the sensor display you're looking at the rear end of your ship). At most stations, there's some buildings with windows and antennae that you'll be facing.
- Velocity not too high (I think the little thing to the left of the sensor display shows this)
From your description, it sounds like you don't have the ship facing "forward" on the pad. You can yaw left or right to get that lined up right, but it will probably be faster to thrust up a bit, pitch 90° up, roll so that you're looking at the top of your ship, pitch back down to level and thrust back down onto the pad.
Sometimes you'll request docking at an outpost and a pad will light up and you'll go for it, but it's not actually your assigned pad. You need to look at the number over the pad and the pad assignment over your sensor display. On the upper left of your sensor display is a small circle with a blue dot in it that can lead you to the correct pad if it's not immediately visible. Just like when navigating to any other location, the blue dot is solid when you're facing the correct pad, a blue circle when you're facing away, and in the center of the larger circle when you're facing the correct spot.
Every once in a while there's a bug where the docking assistance stuff just won't pop up. Or maybe it's not exactly a bug, but a bit of the game designers messing with you. It does seem most likely at outposts; maybe they need to maintain their pads better. Sometimes I've been able to get lined up right anyways without that assistance based on eyeballing the usual landing spot and maybe scraping a little forward and back. It can usually be resolved by exiting the station (or pulling a bit away from the outpost), cancelling your docking request, requesting docking, and trying again. You'll probably get a different pad assignment.
Best Answer
Based on what I've learned while researching how to boost a minor faction's influence, and some reasonable assumptions...
I believe that selling exploration data increases your reputation with the minor faction that owns that station, and also boosts that minor faction's influence. I believe the amount it boosts your reputation (and their influence) will be based on the amount they pay for that exploration data. Since a detailed scan pays more, that will boost reputation and influence more.
For influence boosting, it's believed that the influence boost maxes out at about 10MCr per day per minor faction. It's likely that the same limit applies to boosting your reputation with them. If you're looking to boost your reputation with a single minor faction, it might be best to sell them 10MCr per day until you're done. If you're looking to boost your reputation with a major faction (Federation, Empire, Alliance), you might want to sell your exploration data 10MCr at a time to different minor factions allied to that major faction.
Note that exactly how reputation and influence work has changed in prior updates, and could change again. These changes could be in a server-side patch that you aren't even aware of. Trying to get too detailed with exactly how reputation and influence work can be pretty tricky because of this.