Can powers that grant a basic attack, like many of the Warlord powers, be used to grant an attack to a summoned creature? If so, what does the summoned creature use for its basic attack?
Inspired by Conjuration, Summoning, Pet rules?
[RPG] Can powers grant summoned creatures a “Basic Attack”
dnd-4epowerssummoning
Related Solutions
Summons is now its own effect type with general rules surrounding it.
Pg 221 has the info on Summons that you are looking for in the Player's Handbook 2
Pg 218 has the same info in the Player's Handbook 3
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.
Best Answer
The summoned creature is a valid target of such effects however it has no inherent ability to use basic attacks so the end result is not favorable for you.
The attacks allowed to each summoned creature are as listed for each power. Currently I am aware of none that grant basic attacks. Summoned creatures are not granted any basic attacks by default. Basic attacks are indicated like this
for Melee basic attacks and like this 
for Ranged basic attacks.
The Rules Compendium states on p120 in the Commanding the Creature section that, "The Summoned Creature has no actions of its own...The Summoning power determines the special commands that the summoned creature can receive and gives an action type for each command."
Compare Basic Attacks to Opportunity Attacks.
Opportunity Attacks, like Basic Attacks (as Simon notes) are also granted to all creatures (RC p246). There are many summoned creatures that are explicitly granted Opportunity attacks. See Phantasmist Stalker for one example, its Opportunity Attack is identical to its normal attack.
From this you can infer that Summoned Creatures do not have Opportunity Attacks by default. Hence they should have no Basic Attacks as well.