The Sage Advice Compendium includes a ruling on the DM (not the player) deciding what is summoned with a Summon X spell, and also includes a ruling on dispelling such effects once created:
Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend.
[...]
In contrast, a spell like conjure woodland beings has a non-instantaneous duration, which means its creations can be ended by dispel magic and they temporarily disappear within an antimagic field.
Notice here (backed up by Mearls as asked by @Christopher) that they are saying the creations of a Summon X spell can be ended by dispel magic or temporarily vanish inside antimagic field (because they ARE magic) individually, but that one cast of dispel magic does not destroy them ALL. Targeting the caster with dispel magic would do nothing because the spell is not affecting him as a target; he is simply channeling the spell.
Breaking concentration would be the fastest way to deal with multiple summoned creatures since dispel magic doesn't have an AoE. The aforementioned antimagic field would also be effective if the caster was caught inside, rendering his concentration spell non-functioning in its entirety until he stepped outside of the antimagic zone again, in which case the creatures would reappear.
Any creature created by magic with a non-instantaneous duration disappears
The official D&D Sage advice compendium contains the following clarification:
Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend.
The clarification continues to explain an example of this at play: the spell conjure woodland beings. Conjure Woodland Beings has a non-instantaneous duration, so the magic that summons the creatures is acting upon them and maintaining their presence for the whole duration of the spell. When they enter an anti-magic field, the magic that is keeping the summoned creatures in place is suspended, so the creatures disappear.
When this situation occurs, ask yourself whether or not the effect that summoned the creature is instantaneous. If it's instantaneous, then the creature will stay, otherwise it will disappear while its space is within the field.
To address your examples
Elementals and fiends will only disappear if the spell that summoned them is not instantaneous. A spell like conjure elemental falls into this category. However, a spell like planar ally can summon an elemental or fiend and has a duration of instantaneous. So a creature summoned by that spell will not disappear.
In general undead and constructs follow the same rule as any other creature. Unless the magic that created it is non-instantaneous, a construct or undead can enter an antimagic field with no ill effect (barring any specific exceptions from some other source: like that creature's stat block).
Best Answer
Well, let's take the simple approach to that.
Antimagic-field states that:
Conjuration being divided into several forms, let's look at the subschools just for the sake of precision:
Creation : irrelevant
Healing : irrelevant
Summoning
Hence, it highly depends how you actually bring the outsider to you. Given how you phrased the question, I suppose you referred to the summoning branch of the Conjuration school.
Any summoned creature placing an antimagic-field on itself would have to succeed an caster level check against its eventual spell resistance to make itself wink out of the plane. If it enters willfully in the field, it just winks out1 (until the antimagic-field expires or the summoning duration runs out, and is handled as normal).
While this is not clearly ruled, I'd argue the creature "winks out of existence" but does not disappear as they would if the summoning expired (since it would reappear on the same spot if the field were to go away and the duration of the summoning continue to run as normal), meaning the spells and SLA they casted would remain until the summoning expires. The creature is therefore not stuck in an infinite loop of contradictions, and is just "not there". Personally, since the creature owns the spell, I'd allow it to release it on their turn, as not to spend the duration of the antimagic-field/summoning (whichever is shorter) blinked out.
Any called creature2 or teleported creature3 would be just fine inside an antimagic field.
1 : I hope something powerful enough to use antimagic-field as an SLA would have SR.
2 : Only way I can think of from the top of my head would be using its true name - can't remember which spell actually call a creature... Planar binding or something?
3 : I'm sure you can figure some shenanigans to teleport an outsider on your plane with Teleportation. Probably involves fetching it yourself.