I cannot seem to begin or find this contract: the Phantom of Eldberg, or find it on the notice board no matter how many times I look because it is required before you can begin many other quests, so how do I begin this quest and is something required to begin it that is stopping me from starting it, and why is it not showing on the notice board,
Does anyone know how to solve this problem?

Triggered it just by going to lighthouse area.Then it tells you to check the notice board.

I've been to the lighthouse several times and nothing like that came up to tell me to go check the notice board. Is there a specific place where that happened around the Lighthouse area?

Are you checking the correct notice board? You will (should) find it on the notice board by the Arinbjorn Inn. If not, go inside the inn and talk to Jorund, who posted the contract.

Well it's supposed to tell you when you enter the area at the lightpost about the quest.
You could try Hickory's suggestion or load another save and try that.If it still fails then
dare I say possible broken save or bug.(but not for everyone)which leads me to think
former.

I have the same issue. Just because everyone doesn't suffer from it, doesn't mean it is not a bug. I've been to the lighthouse previously, the quest hasn't triggered, there is no mist in the area, and Jorund is not in the inn, day or night, and I read all the notice board 10s of hours ago.

It's also a fact that many people who complain about bugs (that others don't see) don't, at the same time, also tell you of the mods and/or console commands they have been using.

So what, that doesn't disprove my point. I have never encountered or heard of a bug, at least on PC, that affects everyone playing on that platform, ever. And I am clearly not the only one having issues with this quest, am I? Now, I am not saying it isn't due to a mod I'm using (I'm using about a half dozen), but it's definitely not something I (we) screwed up via console because I haven't used the console in this game yet.

Nor is your point any more valid, ie. that it's a game bug. If you use mods and/or console, you cannot blame the glitch you are encountering on the game, at least not categorically, and certainly since most people, who do not use mods, do *not* encounter the same bug. I've played this game through 3 times, from beginning to end, and have never come across this bug, and it's not just me. With the millions of people playing it, it would be commonly known, but it is not.