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As you know, the buildings in the Operation: Icestorm box come with a wall running around the roofs. This has very often lead to very counter-intuitive situations in demo games and I feel that the rules are not specific enough and do not come with the specific examples required to get the rules right despite what seems logical. What I always explained is this:

1. Walls on roofs (or other scenery elements) are 4 (or more) single pieces of terrain for the purposes of defining Partial Cover/LoF.

I.e. if you are standing on a rooftop, in basecontact to the left wall, you still get no Partial Cover bonus when someone is shooting at you from the right side, the back side or the front side, even though you are a.) touching the scenery element "wall" and b.) less than two thirds of your miniature/silhouette can be seen from the attackers point of view (unless you are in Total Cover from one of these angles, of course).

Is my understanding here correct or have I gone around telling BS?

If so, then the Cover rules are a bit lacking in the specification of the direction of Cover/LoF. They are not wrong and also in themselves coherent but they should stress more the LoF.
The first and second requirement should maybe specify "Partial Cover from the attacking model". I.e. it is not enough to touch any part of the wall going around the roof top, it needs to be the one facing the attacker.

Also, the terrain rules should have included a specification of scenery along the lines of "For the assessment of Partial Cover, a scenery item can only grant cover if it stands between the target and the attacking model" , including a picture example of said situation.

What are your thoughts?

I've just started with Operation Icestorm and agree that some rules are not clear and need interpretation.

My interpretation of cover is the same as yours. You need to be:
a) touching the cover between you and the opponent
b) a third covered

Other rules I had to interpret is if you move for the second part of an order can they ARO in response? And if so can could that model ARO twice? And can you do nothing for part of an order? We interpreted yes/no/yes, though the starter rules as written do not allow this.

Yes, you need to be touching the cover element between you and the shooter. Just having it between you doesn't count. It can be a little counter-intuitive, but you get used to it.

As for AROs, you get an ARO in response to an order, not to each individual skill used as part of that order. So only one ARO.

The way it works:

Active player announces they're spending an order on a model, and announces the first skill being used, and the movement path/target(s).

The Inactive player then announces any AROs from models with LOF or in their ZoC.

The Active player then announces their second skill (assuming the first wasn't a Long Skill).

All dice rolls are made - Normal or Face-to-face as appropriate.

There is an exception for the case where an Active model performs its first skill out of LOF to an enemy and then moves into LOF using its second. That's the only time you can wait until the second skill is announced before declaring AROs.

So, going by your example:

You declare your model is doing something (BS attack, for example, and specify the target(s))

I then declare that these models here will get an ARO. the model you're targeting with a BS Attack may Dodge or return fire, for example.

You then declare your second action will be to Move, and point out the path of that Move (in practice, most people just move the miniature; it's easier that way).
Any of my models that could already see your model don't get a second ARO.

All dice rolls are made. If, say, your model A was shooting at my models X, Y and I had declared that X was Dodging, Y and Z were shooting back as AROS, it would break down as follows:

FtF roll: A's BS Attack vs X's Dodge (BS vs PH)
FtF roll: A's BS Attack vs Y's BS Attack (BS vs BS)
Normal roll: Z's BS Attack (BS)

Note that A can choose to fire from any point along his move (although all shots have to come from the same point) and Y and Z can return fire at any other point; the rolls are still FtF even if the active shots and return fire are taken at different points. For example, A may choose to fire at the end of its move when it has entered a favourable range band, while Y and Z may choose to fire at the midpoint of the move while A is out of cover. Regardless of what happens, the full action is always completed - if A gets shot, they will go Unconscious at the endpoint of the move.

In practice, if you intend to carry out a Move and anything else with a model, you're better off declaring the Move skill first, just to keep your opponent guessing; if you declare the BS attack first, then any models not targeted can make BS attacks using Normal rolls, whereas if you declare Move first, they might choose to Dodge just in case.