Because the celestial eagle summoned by the celestial commander understands him,

Yes, the Eagle Can Aid Another in Combat

According to the spell summon monster I et. al. the summoned creature "attacks your opponents to the best of its ability. If you can communicate with the creature, you can direct it not to attack, to attack particular enemies, or to perform other actions."

The extraordinary ability divine tongue allows the celestial commander to communicate with the summoned creature, no matter its Intelligence score. Therefore the summoned creature can, according to the spell summon monster I, be commanded to aid another in combat without the celestial commander needing to resort to a Handle Animal skill check (DC 25 or DC 27) to push the animal to perform a trick it doesn't know (e.g. the trick aid), which is what other casters must do when they want summoned, unintelligent animals to do their bidding when their bidding isn't biting baddies.

The Secrets of Adventuring
If you're going to use that archetype, and you haven't already, I encourage you to purchase Rite Publishing's Secrets of Adventuring, not because I have a relationship with the company (because I don't have any relationship with Rite Publishing), but because the text might provide further information about the archetype--and maybe even that ability in particular--that's not contained on d20PFSRD.

Yes

As the text on Conjuration (Summoning) says:

bring manifestations of objects, creatures, or forms of energy to you (summoning);

They are not real things, and while that specific manifestation might be wounded, stressed or even dying, you can simply summon another manifestation of the same type of creature using another spell.

In theory, each casting of Summon Monster conjures a different creature from the last, otherwise, it would be impossible to summon 1d3 creatures of the same type.

James Jacobs said:

On Golarion, if you use a calling spell to conjure an outsider, and then kill it, it dies as surely as if you killed it on its home plane. If you instead use a summon spell to conjure an outsider, the thing you summon isn't real before and after the summon spell ends. It doesn't "go back" to an outer plane when you kill it or dismiss it or the spell ends... it just stops existing, just as it didn't exist before you cast the spell in the first place.

But that's on Golarion, Paizo's Campaign Setting, and might or might not be true on other campaign settings.

So, where the summons come from, what happens to them, what they eat and their personalities are explicitly left open for each GM to decide how to handle these things.

Most people treat summons like creatures made out of magic, that only exist while the spell lasts, while others have their own ideas for their home campaigns.

The only known way to summon specific creatures is knowing their True Name (from Ultimate Magic). The rules for that, however, are a little mixed up between Calling and Summoning outsiders, as they give specific rules to enhance Calling spells, but nothing is given about Summon spells.

But we can assume that the general idea works for both subschools, as no exceptions were given to Summoning spells.

So, for a generic setting, we can assume that you cannot summon a specific creature again (say, Bob the Eagle), if it died while being summoned. But you can summon any other creature of that same type.

Rules as Intended

James Jacobs also said, later, on the same topic (yes, it's a long topic):

When you summon a creature using summon monster or summon natures ally, how does it work? Does it conjure a likeness of that creature to fight for you or does it bring a real creature from somewhere?

It summons a "copy" of an idealized incarnation of the creature. A summoned creature doesn't exist before you cast the spell, nor does it exist once the spell expires.

That's the difference between summoning spells and calling spells. Calling spells DO conjure a real creature.

Note that, this time, he did not refer to Golarion when answering the question. Although he could be talking about Golarion when answering this. But this means that the intent of the spell is that you create a creature made of magic using summon spells, not real creatures.

So, even knowing the true name of a creature, it cannot be summoned with Summon Monster.