I noticed that you can't increase your aera reputation by trading nor by flying cargo missions for them. So how do I get them to like/respect me so that they'll eventually stop hunting me?
To get faction from cargo runs, you have to take off or land at one of their facilities. Bit tough when everything in the system is trying to kill you, and faction through cargo is slow.
So you have to kill. What? That is easy and hard at the same time. You kill anything that is not Aera. Of course, this will tend to irritate other factions. Pirates will lower ULN and ULN Citizen Factions. These are easy to raise again as long as they are not too low, you just spam their ships with a "I am a friend" message. F1 when communicating. If they kamakazi you, reload to before they did. You can kill Luddites, this somewhat upsets the purist, but if you kill a few pirates to balance, not a big problem. You can kill ISO. The DGN get mildly upset about you doing so, but again, Pirates will help balance this back.
I'm selling them AI-Cores so why on earth do they attack me in the first place?
You are not Aera and thus are in their list of targets until you raise your faction. It doesn't matter what you are selling them.
I can't go anywhere without aera fighters harassing me en masse which in most cases turns out fatal for me. So what kind of ship can you recommend me? I'm having a hard time in my lancelot, but it's not any better inside a pacifier or franklin. Recently I bought an areus but I'm appalled by its speed and the ridiculously oversized cockpit interior. (Please note that I can't fight them because there's just too many of them for my graphics hardware to handle.)
No player ship is going to stand up. You also cannot fight them because killing them seriously reduces your faction. For the graphics problem, change your Video, shaders, resolution and colors to the minimums. Yeah, it looks crappy, I know. Also use full screen instead of windowed and turn sound off. Sound uses up cpu cycles that you want for video. If you are using Linux, custom build you a Kernel that uses only the bare minimum needed. You could then boot into it when you want to play. For Windows and Mac, don't know a lot that will help. Just use the most current driver and minimise the amount of apps that are running.
They somehow offer cloaking devices but I fail to purchase one: no ship I tried (including the aera ones) could take it. So how do I get it?
You have to have a stealth capable ship. Check out the Wiki to see if there is more info on it. I think the Shroedinger (sp?) is supposed to be stealth capable. You can check the units_working.sxc in the VegaStrike/data/units folder. It will tell you which ships are stealth capable.
Is there a feasible way to (a) do small military jobs for the aera while (b) not pissing off most other factions?
I assume you mean bounty mission, sweeps, etc, aka, you are killing something. Nope, you kill something, you upset their faction and the faction of their friends. If you get the factions balanced the way you want, doing Cargo is the only thing that is fairly safe. I do not know if doing cargo runs for them will have negative influence on other factions, just keep a watch on them.
Anything else I should know about the aera?
They hate everyone and everyone hates them. Get used to other factions doing mass attacks on their systems. Hopefully you are not negative with the faction doing the attack, otherwise, run fast and pray.