In the future, please ask separate but related questions separately.
When a bound creature is controlled by another being, does the binding end?
No, binding is a spiritual bond between the creature and its prison, accompanied by the frozen spiritual state. Binding is linked by a spiritual bond to the summoner who performed the binding (through the Zeon upkeep) and can't be transferred unless the bond is transferred somehow (e.g. by any of the methods of transferring spell upkeep) which Control explicitly can't do. This is extracted from the descriptions of the various summoning skills in specific and the "Opposing Abilities" section on pg 177-178 of the core rulebook (I have the English version. Page numbers will be off for the original Spanish).
Can a binder return a bound entity to confinement if he or she loses control?
Yes, but it doesn't return control of the entity to the binder. While the entity still has a frozen spiritual state (and thus can't try to break free of either the binding nor its new master) and, more importantly, can't anything but communicate while bound (which might still be a big deal. I'd give a binder negative penalties to his secondary abilities if he had to do stuff--like combat-- while the demon bound in his staff screamed constantly into his mind.), it still remains controlled by the opposing summoner and if Alice brings the demon back out, it's still controlled by Bob.
This comes from the same area (pg. 177-178), and relies heavily on the passage stating that Control can only be used to oppose Control (and thus can't interfere with Bind and vice versa). The various penalties for binding are taken to support the position that Binding is relational, and not just a property of the bound being. Other interpretations are possible, depending on your philosophical answers to the questions of the metaphysics of the soul.
Who gets control when the person controlling a summon dies?
No one for a few hours (1 hour per point of Power of the Summoner). This comes from the 'Called Away' section of the 'Between Life and Death' Chapter (Chapter 7), and the Breaking the Bond section (which is specifically about familiars, but the Familiar Powers section on the same page indicates that bound and controlled creatures function similarly) on page 182. I also generally rule that all spiritual bonds break on death unless they specifically say otherwise, and this rule has served me well in my GMing.
After that the summon is released if bound (binding is a spiritual bond) and has control of itself until someone else uses Control on it. Only one person can have Control of an entity at a time (see the Opposing Abilities section again); former summoners are not (and cannot be) 'waiting' to resume control.
What happens if Alice's bound summon is controlled by Bob, who she kills?
The summon is still bound, but no longer controlled. Control must be reestablished, which requires a new Control check and can only be done while the summon is outside its prison. It should be noted that this may be very difficult as the being, once released from its prison, will almost certainly attack the would-be Controller. The being can be contained by the binder indefinitely without being controlled, however (assuming upkeep is payed, etc). See the above sections and, again, pages 177-178. The disposition of the previously enslaved being is based off of common sense and the failure table on page 180.
What happens if Bob successfully takes control of one of Alice's non-bound summons and then releases it?
It's free. It probably doesn't attack Bob, at the very least, and may offer something in the way of thanks dependent on its nature. It likely flees, given Alice's ability to Control it in the past, but may fight Alice depending on its abilities and inclinations. See the above section that has to do with Alice dying for rule sources. Summon behavior is just what I think most likely, it does whatever makes sense for it as an NPC. In really dark games that might include Stockholm Syndrome.
What about Banishment?
Banishment doesn't end control or binding, but it does send a creature home and forcibly prevent its return, usually for a very long time. The societies of most creatures that are commonly summoned probably have methods of dealing with this, and it is extremely unlikely that the creature will remain bound or controlled for any length of time. If bound, the summoner cannot force the creature into its prison during this time, unless that prison first leaves the material world (a very reasonable alternative ruling would be to have Bob's banish oppose Alice's bind when she makes an attempt to do so. The failure table on pg 180 makes such a confrontation very dramatic, however, and very biased towards Bob). Alice can't see what's happening to the summon once it's gone, so sending orders through the link in an attempt to forestall its being freed by its society is unlikely to be effective, though attempts can be made and Alice does get to learn the approximate location of the summon's home. This is combination of the text on pg. 177-178, Chapter 20 (which describes very briefly and vaguely the Wake, one of two places summoned beings can come from), and the Heavenly Essence ability on pg 282 (which mentions some Maidens of Light, a kind of elemental, as having a "close relationship with Mikael", who is one of the most powerful beings in existence. It supports the idea of elementals living in societies and working together).
What about Familiars?
You get a familiar by binding a being to you. Like, inside your soul. Obviously this has some drawbacks. See pg 180-181.
The main advantage to a 'Union Bond' is that, unlike normal bound beings, the familiar never has to (and indeed, can't) leave it's prison and so can't be subject to further summoning abilities, as if it were inside the object it were bound to. This a consequence of part of your soul being cut off and placed in it, just as part of its "blood, flesh, and spirit" is placed in you. As a result, as long as the bond persists enough of its being is within you, the object of the binding, that it is protected. See pages 180-181, especially 'Union Bond' and 'Breaking the Bond'.
But what if it happens anyways, like with magic?
If the familiar is banished somehow (presumably not by the banishing skill), the master can still see through its senses as if he or she were there, and so has a much better chance of getting the familiar back or to a safe waiting place un-freed. The summoner can also cast spells directly through the creature, which is likely to make its banishment moot for summoner/spellcasters. (see the various spells that cross worlds or increase movement enough to let someone do so, and 'Familiar Powers' section on pg 181.
What about using magic on non-familiar summons?
Well, first of all, we need to know what Bob is targeting. Usually he's going to be targeting the manifestation of the being in the material world (like with Fire Ball or Return to the Flow) in which case the creature uses its MR (which the summoner should really be buffing since that protects against hostile summoners as well). If he tries to control it he may be targeting its mind, injecting or transmigrating some or all of his soul into it, or using some other method made moot by the Control skill. If he does this successfully, he gains control of the portions of the creature that the spell says, but controlling those portions doesn't actually give him any influence over the creature's actions (and might even be dangerous to him or lead to him being controlled, like with Transmigrate Soul). If he's controlling the creature's body, like with Atomic Control (which would work on Earth elementals but not Fire elementals), or controlling its future (like with Predestination) the summoning is made moot. If he uses Supernatural Control or other similar spells against the Control secondary ability in particular, the rules for Opposing Abilities (again on pg 177) apply and "the character with the highest Final Ability Score overrides his opponents Summoning attempt. If, for example, two characters attempt to Control the same entity, the character with the highest result will dominate the being". The relevant ability for the spellcaster is Magic Projection, the relevant ability for the Summoner is Control (note that while control can only be self-opposed within the summoning abilities, other abilities, like magic, can contest it freely). Some Magic Projection stuff can be found on pg 110.
When spells specify that the highest MR involved is used, the higher of Alice's and the creature's MR is used. If what is actually targeted is Alice, or Alice's soul, or a fragment of Alice's soul, or something similar Alice's MR is used. If what is targeted is the summoned being, that being's MR is used. The general rules for MR can be found on page 14-15 but they aren't very helpful.
Contested Orders
As long as the Summoning abilities are in use, this will never come up. Only one person can control the being at a time. See pg 177, one last time.
