Spells can be cast through the Antimagic Field
I agree completely with DaleM's answer but thought I might provide a different way of getting to the same answer.
Antimagic field does not block or change the targeting rules
There is nothing in the spell effect description that says that antimagic field has any effect on the ability to target things magically or mundanely through it. In fact, antimagic field does not even have any effect on the ability to target things inside of the AMF.
Targeted Effects: Spells and other magical effects, such as Magic Missile and Charm Person, that target a creature or an object in the Sphere have no effect on that target.
The only change antimagic field makes is that creatures targeted with magical effects while inside of it are not affected by them.
Thus, the normal rules of targeting apply and antimagic field does not change them at all.
Magical effects crossing into the AMF are suppressed, but not dispelled or blocked
Spells and other magical effects [...] 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.
In this case, "suppressed" is being used to mean that the effect stops working while it is in the field. Note specifically though that it does not dispel or block any effect. The effects simply stop working in the AMF. Spell durations still count, concentration still has to be maintained, etc. If the AMF actually blocked or dispelled these things, the spell description would say that instead.
So, to take a simple example, if I were to cast light to create a magic effect on a rock in my hand and walk into an AMF, it would stop glowing. However, as I came out the other side of the AMF it would immediately light up again (assuming I am still within the duration of the spell).
That same principle applies to any magical effect passing through the AMF.
So, fireball for example would, upon being cast, essentially travel from the caster's finger through the AMF (becoming invisible in this area) and then becoming visible again before hitting it's target on the other side of the AMF. Of course, because fireball is an AOE effect it is effected by another clause of antimagic field's description:
Areas of Magic: The area of another spell or magical effect, such as Fireball, can't extend into the Sphere. If the Sphere overlaps an area of magic, the part of the area that is covered by the Sphere is suppressed. For example, the flames created by a Wall of Fire are suppressed within the Sphere, creating a gap in the wall if the overlap is large enough.
So if the actual fireball that results from hitting its target overlaps with the area of AMF then there is no fire in that area.
And this is explicitly according to the rules laid out in the description of antimagic field.
Applying this to your specific question yields the following:
- Magic missile would hit.
- Fireball would hit its intended target (and any AOE overlap with AMF would be suppressed)
- Cone of cold would be cast as expected (any AOE overlap with AMF would be suppressed)
- Polymorph takes effect if the creature meets the conditions in the spell (failed saving throw or willing)