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.
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
It would scale with the creature casting the spell
The term "radius" is usually defined the distance from the center of a circle to its boundary, which would make this spell seem at first to create a "globe" of fixed size. But there is some language in the spell that gives us a hint that the term "radius" is intended in an atypical way. Consider the wording of Globe of Invulnerability (PHB, p. 245, bold and italics added):
The not only are we told the radius, but that it is a radius "around" you. That is highly unusual terminology: is it meant to indicate that the radius begins at your outer boundary and thus it scales with your size? Or is the term "around" simply meant to complete the previous phrase ("a... shimmering barrier springs into existence... around you"), and this language was included just based on the assumption of a small or medium (PC) caster? In isolation, there's no way to tell.
Similar wording elsewhere shows us the way
The terminology of a "radius around" a creature is used infrequently in the PHB, and some of those uses (such as in the description of the spell Holy Aura) are equally ambiguous. However, there is another place where this terminology is used in the PHB where its meaning can much more clearly be derived. It is found in the rules on Creature Size and Space (PHB, p. 191, bold and italics added):
The terminology is grammatically identical to the terminology used in Globe of Invulnerability, except for the "radius" being 5 feet instead of 10. So we will examine what they meant in this case, which will give us guidance on what they meant in the other.
Consider the following image, where the squares represent 5 foot squares, the center of which contains a creature.
The orange and purple curves represent the two possible spaces defined by a "5 foot radius around" the center square. The orange curve assumes that the "around" part means that the distance starts at the edge of the central square's boundary. The purple curve assumes that the "5 foot radius" is from a central point in the center of the 5 foot square.
Neither the purple nor the orange curve completely cover the eight squares around the central square, and both cover some part of each of them. But the Dungeon Master's Guide give us guidance for when to include an area "in" a radius: specifically if an area covers "at least half a square" (DMG, p. 251). We can see that the purple curve covers half of none of the outer eight squares (certainly not the four squares in the corners). But the orange curve covers half or more of all of the eight square surrounding the central one. And since we are told that "Assuming Medium combatants, eight creatures can fit in a 5-foot radius around another one", it seems likely that "radius around" means the distance begins at the outer boundary of the creature in the center, not at its central point.
Auras help as well
There is further evidence of this conclusion in the DMG. The term "radius around" is used infrequently there as well, specifically to refer to features of the Oathbreaker paladin and the Holy Avenger weapon, which can only be used by Paladins (p. 97 and p. 174 respectively, bold and italics added). In both cases, these terms refer to features of a Paladin's Aura.
A paladin's aura is a key example of a feature whose boundaries are defined not by their distance from a single point but rather from the outside of a Paladin's space. In the case of other Paladin aura features, the language is much clearer on this point (such as the Aura of Protection which affects creatures "10 feet of you"), and contains none of the ambiguity found in the "radius around you" phrase. However, there are several hints that the two descriptions of auras should probably be treated identically.
The distances provided in these two features are the same as the distances given other paladin auras (10 and 30 feet respectively). They also scale similarly, in one case starting at 10 feet at lower levels and rising to 30 feet later. To have a Paladin's usual auras (defined as working for creatures "x feet from you") work differently than these auras in the DMG (defined as working in a "x feet radius around you") would create a complicated situation where several of a paladin's auras are one size and others are 2.5 feet smaller in radius. This complication gives us no tangible benefit, and could confuse and slow play in many situations. It would also confuse matters when played on a grid. As previously stated, the DMG states that a circular area of effect is only used on a square if the area covers half or more of a square. In the case of a 10 foot radius centered at your center, this radius would not cover half of any square that is more than 5 feet away from the paladin. Thus, it would effectively be an aura that only covers 5 feet "from you". This point of confusion is highly unlikely to be intentional, so it appears more likely that the two terms ("distance from you" and "radius around you") are meant to be used interchangeably.
Jeremy Crawford (unofficially) Agrees
The rules on a spell's "point of origin" are peculiar: they state (PHB, p. 204):
Note that they do not say that the point of origin is "on" a creature or an object, but rather that the creature or object is the point of origin. It is unclear what is meant by this: is the point of origin still a "point" located in the center of the creature/object? Or do the rules on "point of origin" sometimes refer to something larger than a "point"?
Jeremy Crawford has specified this in an (unofficial) conversation on twitter:
Destructive Wave has a range of "Self(30 foot radius)", while Globe of Invulnerability has a range of "Self(10 foot radius)". From Crawford's answer, we can understand that any spell with a range of "Self(X foot radius)" uses the creature (or more specifically the creature's space) as a "point of origin". And we can see that the "point of origin" from which the radius of the spell extends is no longer a single (zero dimensional) location, but rather the entire space of the creature.
Ease of use
The reality is that it is simply easier to use this interpretation in most cases. If a creature has a 5 foot reach, and another has an effect with a range of "Self(5 foot radius)", you'd expect both creatures to be within each other's respective ranges when they were both five feet from each other. Having to account for the thickness of a creature's space or the radius from the center of its body complicates the situation needlessly.
Putting all these features together, we can see that the Globe of Invulnerability spell will scale up with the size of its original caster. Depending on whether or not you follow Jeremy Crawford's advice, it may scale with that creature's space, or perhaps with its body. It would probably be easiest to determine the distance scaled from a creature's space, as a creature's body (especially for an elongated creature like a dragon) can be difficult to map out quickly. Either way, the caster will invariably be inside the Globe when it is cast, and the globe will invariably begin "around" the caster.