You both roll.
Perception is a skill, not a sense. Two people see exactly the same thing, and one of them notices a small detail or something out of place that the other does not notice. (Sherlock Holmes: "You see, but you do not observe. The distinction is clear.") When your own perception skill increases with more levels, it is not because you magically grow a better pair of eyes. You learn how better to utilize whatever eyes you are looking through.
You and your familiar are two different creatures looking at the same picture. You both can apply your observational skills to that picture and tell each other what you notice.
Antimagic field only suppresses areas of effect while they overlap with the field; they function as normal otherwise
The antimagic field spell description answers your question:
A 10-foot-radius invisible sphere of antimagic surrounds you. This area is divorced from the magical energy that suffuses the multiverse. Within the sphere, spells can't be cast, summoned creatures disappear, and even magic items become mundane. Until the spell ends, the sphere moves with you, centered on you.
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.
[...]
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.
As the phrasing of the sentence I've bolded suggests, the area of effect of other spells and magical effects is suppressed only while they overlap with the antimagic field.
Those parts of an AoE outside the area of antimagic field are not suppressed (though I'm not sure how it works if the point of origin of a spell like darkness is within the area of antimagic field). Once either the other AoE is moved or you move (since antimagic field is centered on you) such that there is no overlap, there is no AoE in the field to suppress.
This interpretation is further supported by the spell's description of its interaction with creatures/objects summoned or created by magic:
Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.
As you can see, antimagic field doesn't dispel/end the other spell; it simply "suppresses" its effects while the field overlaps the effect of the other spell (or magical effect).
Best Answer
To answer your last question first, you are correct, a familiar is not an ongoing magical effect. It is a supernatural being, but it wouldn't show up to a detect magic spell for the same reason a dragon, zombie, or elemental wouldn't. The Sage Advice Compendium question that impinges on this situation is a bit long, so I'll reproduce the text here, and focus directly on familiars at the end.
So we'll ask ourselves those questions about familiars.
While the qualities you listed are supernatural, none of them are magic items or mention specific spells, and none of them are explicitly magical within the description. The touch-spell delivery is specifically using a spell (and acts 'as if the familiar had cast it'). However, in the Monster Manual (page 9) and the corresponding section of the basic rules, telepathy is described as 'a magical ability', and more to the point, the last paragraph of the description of telepathy has this to say:
So, in an antimagic field, the familiar's ability to channel your touch spells and its telepathic link to you would be disabled, but all its other qualities, while supernatural and amazing, are not magical game effects any more than a dragon's breath is, and would not be disabled. Your DM may decide that the sense-sharing ability is an advanced form of telepathy and would therefore be blocked as well, but it isn't explicitly so as far as the rules as written.
In the case of other effects that block 'magical effects', such as Leomund's tiny hut:
Spell-channeling wouldn't pass through a barrier that blocks magic, as it's actually casting a spell (bullets #2 and potentially #4), and while telepathy is only explicitly blocked by an antimagic field, the description does specifically call it out as a magical ability, so it falls under bullet #5 and would be blocked.
The other familiar abilities would probably not be hampered since they are supernatural but not magical in the aforementioned sense of 'concentrated magical energy that is contained or channeled'.