[RPG] How do the cover rules interact with the Flaming Sphere spell

No.

The power's description doesn't explicitly mention the Melee keyword for the sphere's attack, therefore that attack is not a Melee attack.

The only restriction is that the attack's target has to be adjacent to the sphere, but a close blast 1 or a [non-provoking] ranged 1 attack has the same effect. This means that the range restriction cannot be used to guess any implicit keywords of the attack in that way.

Compare the wording of these Conjuration powers, some of which explicitly mention the proxy's attack's type:

Flaming Sphere^[DDI]:

[...] You conjure a Medium flaming sphere that occupies a square within range, and the sphere attacks. [...]

Mordenkainen's Guardian Hound^[DDI]:

[...] You conjure a Medium guardian hound that occupies 1 square within range, and the hound makes the following melee attack. [...]

Ball Lightning^[DDI]:

[...] You conjure a Medium ball of lightning that occupies 1 square within range, and the ball makes the following attack, which is a close burst 1. [...]

The Wizard may Ready an action to cast a spell setting the trigger to be when the orcs have finished moving.

What will occur on the Orcs' turn is they will move (potentially in a cluster), the Wizard will cast the spell, and the Orcs will do something with their Action. It should be noted, they might respond to the spell by using their Action to dash, which would let them run away; they could choose to just attack the Wizard and attempt to break his concentration, or something else.

In addition, your question suggests that you might be confusing a Round and a Turn, which are separate and distinct things from a timekeeping perspective. A Turn is an individual's activities that occur during a Round. A Round is the sum of all the creatures' activities and always equals 6 seconds. Some effects will say that they occur at the beginning or end of a Turn, which refers to the individual creature; other effects will indicate they occur over some number of rounds (most often represented with time increments like 1 minute).

So for your specific issue, the damage is dealt at the end of a creature's turn. This is an individual determination. For example, were there Orcs and Wizards in this fight, the sphere deals damage to the Orcs only if they qualify (are within 5' of the sphere) at the end of the Orcs' turn. At the end of the Wizards' turn, the same assessment is made to determine if the Wizards take damage. The spell also permits an additional effect that allows the Wizards to attack with the sphere more directly during their turn.

In tight spaces, it's possible to deal a lot of damage with this spell passively by virtue of its presence, but in most fights it's probably only dealing damage on the Wizard's turn. Otherwise, it's serving to deny access to an area because most folks don't want to end their turn taking 2d6 fire.

Regardless, if the sphere is still adjacent to any Orcs at the end of their turn, they will suffer the effects as described in the spell.