No, it does not need to be able to see for the initial effect
There is a principle that spells do what they say they do. If the spell does not say the target has to see the source to be affected, then it does not. Fear says:
You project a phantasmal image of a creature's worst fears. Each creature in a 30-foot cone must succeed on a Wisdom saving throw or drop whatever it is holding and become frightened for the duration.
There is no language there about needing to see the caster.
In addition, a phantasm is traditionally an illusion that is created in the mind of the creature, not externally. For example, phantasmal forces says "You craft an illusion that takes root in the mind of a creature". That would also support why they do not need to actually see anything to be affected.
For the other questions:
If they can move 5 feet, do they take the Dash action and stop?
While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move.
If they can move, they always must take the Dash action, there is no exception. They will move as far as the movement from the Dash action and their other movement allows.
Would they have to move their full movement before taking the the Dash action?
The order in the text is that they first take the Dash action, then move, but I do not think it matters.
How much space is needed?
From the wording it is not entirely clear if the "... unless there is nowhwere to move" extends all the way back to the Dash action, or only refers to the movement part of what comes before it. I think that it makes more sense for it to apply only to the movement: it would be narratively weird if they would need to take the Dash action to move 5 feet and therefore could not take other actions1, but they do not need to take the Dash action if there is nowhere to move, and instead can use their action to attack you. But I think this will be up to the DMs interpretation/adjudication.
What if the way isn't safe?
The text does not say they only need to move if there is a safe route. They have to use the "safest available route", that means if all routes are dangerous, then they need to move through harmful territory, as long as they can move there. If the only route leads through a Spike Growth area, that is an available route and also the safest available route, and they will need to take it. They can pick the least harmful, e.g. if they have the choice between a lava stream and a thorn bush, they can take the thorn bush, even if it means a less direct route of getting away.
1 Well, at least not without bonus action tricks