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. [...]

Yes

You are targeted by the effect if you:

• are in the area when it activates
• enter the area for the first time on any turn
• are in the area at the end of your turn.

It doesn't matter if you have previously passed or failed a saving throw or are already under the effect: you are targeted - if you fail the saving throw you suffer the effects (possibly again).

I'll go through each of the effects to explain what it means for each of them:

• Death: 10d10 necrotic damage on a failed save, halved on a successful one each time you are targeted. This effect alone (damage) does not stop it moving out on its turn and not being targeted further.
• Discord: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. This effect alone (special) does not stop it moving out on its turn and not being targeted further.
• Fear: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. This effect (fear plus special) actually forces it to flee the effect so it would only make further saves if, after it recovered, it reentered while the symbol was still active.
• Hopelessness: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. This effect alone (special) does not stop it moving out on its your turn and not being targeted further.
• Insanity: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. Its movement is "erratic" so it may wander in and out of the area several times, if it enters on its turn it must save, if its still there at the end of its turn it must make another save.
• Pain: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. This effect alone (incapacitated) does not stop it moving out on its turn and not being targeted further.
• Sleep: If the creature fails a save it suffers the effect for 10 minutes. If it fails a latter save it suffers the effect for 10 minutes from that save. This effect (unconscious) prevents movement so, unless helped by someone else it is going to be in there for a long time: it will sleep for 10 minutes after the effect ends less 6 seconds for each sequential save it made working backwards from the end of the effect.
• Stunning: If the creature fails a save it suffers the effect for 1 minute. If it fails a latter save it suffers the effect for 1 minute from that save. This effect (stunned) prevents movement so, unless helped by someone else it is going to be in there for a long time: either until it makes 10 saves in a row (unlikely) or until up to a minute after the symbol stops functioning.

If you can keep a creature in the area of a Symbol (Death) for the full minute they will almost certainly die. However, because of the rules on combining magical effects, you can’t use another Symbol to do it.