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.
Conjuration
Each conjuration spell belongs to one of five subschools. Conjurations transport creatures from another plane of existence to your plane (calling); create objects or effects on the spot (creation); heal (healing); bring manifestations of objects, creatures, or forms of energy to you (summoning); or transport creatures or objects over great distances (teleportation). Creatures you conjure usually- but not always- obey your commands
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.
Calling: a calling spell transports a creature from another plane to the plane you are on. The spell grants the creature the one-time ability to return to its plane of origin, although the spell may limit the circumstances under which this is possible. Creatures who are called actually die when they are killed; they do not disappear and reform, as do those brought by a summoning spell (see below). The duration of a calling spell is instantaneous, which means that the called creature can't be dispelled.
Summoning: a summoning spell instantly brings a creature or object to a place you designate. When the spell ends or is dispelled, a summoned creature is instantly sent back to where it came from, but a summoned object is not sent back unless the spell description specifically indicates this. A summoned creature also goes away if it is killed or if its hit points drop to 0 or lower, but it is not really dead. It takes 24 hours for the creature to reform, during which time it can't be summoned again.
When the spell that summoned a creature ends and the creature disappears, all the spells it has cast expire. A summoned creature cannot use any innate summoning abilities it may have.
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:
Creatures summoned using this spell cannot use spells or spell-like abilities that duplicate spells with expensive material components (such as wish).
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.
Components V, S, M (holy or unholy water and incense worth 500 gp), DF
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.
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
Most other features are explicit that they are set at the time of summoning, but the attack modifier is not specified
The aberration has a number of features that are explicitly determined when the spell is cast
The AC, hp, number of attacks, and damage dealt by the aberration are all determined at the time the spell is cast, based on the level of the spell slot that was used.
Further,
Even if you later choose another kind, that will not change its traits.
The aberration has just one aspect that is explicitly based on your current state
Summon Aberration is a concentration spell. If you lose your concentration the aberration will no longer be able to manifest at all.
The aberration has two features that do not explicitly say whether they are determined at the time it is summoned, or are instead based on your current state.
As you mention, the attack modifier of the aberration is determined as "your spell attack modifier". In addition, if the aberration is of the Star Spawn kind, its Whispering Aura has a spell save DC based on your spell save DC. Notably, neither the single qualifying word "current" nor a qualifying phrase such as "at the time you cast this spell" is present in the description. Thus, it is unclear which of the two possibilities determine these features. Your DM will need to decide how to implement these.
Other spells and abilities do implicitly assume 'as the spell is cast' in their descriptions.
This question is about storing spells in a Shield Guardian, which says the spell's parameters are "set by the original caster". This answer to the question explains that this means they are set at the time of casting and don't change as the caster changes.
Both the invisibility and greater invisibility spells include the phrase a "creature you touch becomes invisible until the spell ends. Anything the target is wearing or carrying is invisible as long as it is on the target's person." When I first read this, I was confused, because when I read "Anything the target is wearing or carrying is invisible as long as it is on the target's person." I could not understand what the target could be wearing or carrying that was not on their person. The spells only make sense when you understand the implicit 'at the time of casting' and parse them like so:
What I would do
Narratively, you invest energy in the creation of the form of summoned creatures, or in bringing to your location the most powerful form you can, but once you have finished the summoning, they act as independent creatures under your direction. While you need to maintain your concentration on them to preserve their existence, their form itself has already been determined. It makes no narrative sense for the aberration's physical form to be continually 'checking in' with your state and then adjusting accordingly how well it can attack or how difficult its psychic energy is to resist.
Given the preponderance of features in the summon aberration spell that are explicitly determined at the time the spell is cast, the number of other spells and features that implicitly assume 'at the time of casting' in their descriptions, and in the absence of counter-examples of summoned creatures whose abilities are tied to the current state of their summoner, it seams more reasonable to assume that the two features not explicitly listed are also determined at the time of casting as well.