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.
I am certain that your problem is caused by your Sensors being disabled in the Modules tab of the right hand interface screen. I have tested this myself and got the same results as you.
Open the right hand interface screen, change to the Modules tab and scroll down to locate the status of your Sensors.
If the text for the Sensors entry is yellow, then the check-box on the for them will be unchecked. Toggle it so that it is checked and you should get your sensors back.
If the text for the Sensors entry is red, the check-box will have a line through it as it is disabled due to your Power Plant not providing enough power for all your modules. In this case, what you need to do is change the priorities of each module so that they are available when you need them to be.
Highlight a module, and use left and right to change its priority. A value of 1 means highest priority. I suggest setting all modules that you want to be available always (Thrusters, Shield Generator, Scanner etc) to priority 1. All weapons/combat modules (Kill Warrant Scanners, Cargo Scanners, Point Defence) to 2 and everything else to 3.
With this configuration (assuming your Power Plant can power your weapons when they are deployed) you should be fine. When you deploy your weapons, if you don't have enough power, things like the Cargo Scoop, Fuel Scoop and Frame Shift Drive all shut down but you can't/won't want to use these when fighting anyway.
Note: If a module doesn't consume power, you can't change its priority (such as the Power Plant, Discovery Scanners and Cargo Racks).
Best Answer
Missions are found either while in space (e.g. Distress Calls) or from Bulletin Boards (BB). Since you are new to the game I would advise to stick to BBs for the moment. They display all available missions, whether you meet the requirements to accept them or not. The interface will split missions into "Available" and "Unavailable". If no missions are available to you you can wait until the BB refreshes, typically 10 minutes, and check again.
It may, however, be that no missions show up for you repeatedly. In that case I'd advise to get to a new station. Don't be afraid to visit another station in the same system (costs about 10Cr in fuel) or a nearby system (costs more fuel but is still affordable). You can always take on some cargo and transport it to a system that is noted to receive this cargo as export from your current system. You will be guaranteed to make a profit, even if not a huge one. They should however cover all your travel costs.
What mission you want to take on is up to you and how you want to play the game. Generally, you can take on trading or errant missions, combat missions against pirates or specific non-pirate targets, or mining. There are a few other types that have been added with the latest patch but all of these fall into two main categories as specified by the rank they require; they either require a combat or merchant rank.
As a last advice, don't be afraid to die or run out of fuel (and self-destruct). You will always be able to reacquire your starting ship at no cost (other than the inconvenience of defaulting back to your start location).