While a summoned monster does initially act on the turn of the person summoning, it has its own initiative track. If later it delays its action, or uses a spell or ability that changes its initiative, its initiative changes to a different count than the summoner's (or vice versa).
The rule you're quoting only says that it gets to act on the turn it was summoned, on the same initiative as the summoner, not that their initiative can't change individually later on.
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
Summoned creatures are not real creatures.
It's important to notice that a summoning is not a calling. Although they are both from the Conjuration school, you do not summon a specific creature, you summon a generic specimen of that creature type. The summoned creature cannot be summoned again until 24 hours is passed, while it reforms itself on another plane.
As we can see here, the main difference between a calling and a summoning is that one transports a creature, while the other brings a manifestation of a creature.
We then see further differences between the two types of conjuration, specifically about the manifestation reforming or the creature dying if killed.
We can tell that the rules are a little vague about this. But James Jacobs also seems to support the idea that summoned creatures are not real, but manifestations based on real creatures.
Commune Spell-like Ability from Summoned Creatures
If the creature was summoned via Summon Monster spell, it says this on the spell effect:
We can see on the spell text of Commune that it does have an expensive material component, thus summoned creatures cannot use their Commune spell-like ability. This is because summoning spells do not allow it, not because the creature may or may not have used it this week.
For spells without components, the monster should have all their spell-like abilities available when summoned.
It's possible that there exist Conjuration(Summoning) spells that are not copies of the Summon Monster spell, and those, if their spell text does not say functions like Summon Monster or does not forbid the use of expensive spells, should allow their use. But, as far as I know, no such spells exist. All spells that summon specific creatures, such as Summon Genie or Summon Thanadaemon, also specify that they work like Summon Monster.
There are a few spells that are a Conjuration(Summoning), but does not have the expensive material clause, such as:
Conjuration(Calling) Spells
Calling spells are a whole new deal and allow you to use their spell-like abilities, if that's part of the contract, in some cases, and if the creature likes you enough, in others.
Spells like Plannar Ally, Gate and Planar Binding will conjure real creatures, with names and personalities, and should have all their abilities on their stat blocks available. They might also have things not listed on their bestiary (or source book) such as different equipment (axes instead of swords), different personalities (a neutral succubus?), different feats and skills, and even different classes since many of those creatures are intelligent enough to take class levels.
Example: a conjured Imp can use his Commune spell-like ability.
There might be exceptions to this, but this is the general ruling for calling spells as they have nothing that prevents them from using their spell-like abilities.
But be warned to not abuse those creatures when you conjure them, as you could make terrible enemies that will remember everything you have done to them while they were under your control.