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.
Antimagic Field will prevail
We can look at this from two points of view, both from the rules and from a purely RP perspective. Let's check the rules first.
The description of Antimagic Field states that
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
and that
Spells and magical effects such as dispel magic have no effect on the Sphere. Likewise, the spheres created by different antimagic field spells don't nullify each other
However, the Globe of Invulnerability can be dispelled or suppressed, as it specifies that only what is inside it is protected, and not the globe itself. Therefore it stands to reason that when the two spells meet, Antimagic Field will suppress the Globe of Invulnerability itself, which will allow the field to affect creatures or objects within the globe.
If we look at it from a RP point of view, we can once again turn to Antimagic Field's description, which says of the area of the field:
This area is divorced from the magical energy that suffuses the multiverse.
We could then say that this area lacks the required property to sustain a spell, which would suppress the globe. This isn't a case of a spell being stronger than another, such as light spells vs darkness spells. This is an instance of a spell removing the necessary energy for another spell to sustain itself, thereby making it fail.
Best Answer
Yes, summoned monsters are subject to Globe of Invulnerability, can be dispelled, detect as magical, etc. Do keep in mind that called creatures are not - see the details of the Conjuration sub-schools. If the spell is ongoing, it's simply suppressed while in the globe (the duration continues to go on, and the spell comes back if it moves outside the globe).
Unless, of course, the Summon spell is already in effect when the globe is created, just like every other spell, per the rules of the Globe.
It's not really that confusing, it's just that there are several cases.
Existing - In Place. You create the sphere in an existing spell's area, like a wall of fire on top of you. No effect, the wall of fire is still there.
Existing - You Move. You create the sphere and move it onto some existing spell, like a wall of fire. No effect, the wall of fire is still there.
Existing - They Move. You create the sphere and an existing spell moves into it - a summoned creature or a flaming sphere or whatnot. The spell is not affected and it gets you, as it is a "spell[s] already in effect when the globe is cast".
Newer - In Place. You create the sphere and someone tries to pop a wall of fire or whatnot on top of you. A wall of fire appears with a spherical hole in it. The spell just doesn't function within the sphere.
Newer - You Move. You create the sphere and someone else casts a spell like wall of fire. You move into the wall of fire and the sphere suppresses it within its radius.
Newer - They Move. You create the sphere and then someone casts a spell that moves into it - a summoned creature or a flaming sphere or whatnot. It's suppressed while within the sphere. You can skin that as "can't enter the sphere" or the slightly weirder "it disappears, then the caster moves it back out and it reappears?"
The spell makes no distinction about whether the globe is moved onto a spell or a spell moves into it or whatnot. The net is that any spell preexisting is unaffected and any newer spell can't come into the sphere under any circumstance.
The one end run around this is spells that create a thing that isn't dispellable or whatnot, like some conjuration (calling) spells and instantaneous duration conjuration (creation) spells. So while a wall of fire (evocation) is affected, if a caster creates a wall of iron after you've popped the sphere, it is unaffected by your sphere. This is the standard "conjuration dodge" used to affect creatures with SR, etc. Note that summoned monsters bypass SR simply because the SM spell in particular has "SR:No," but that's a case of specific trumps general. In general, conjuration (summoning) is a magical effect subject to Globe, SR, etc.