Does antimagic field actually prevent the casting of a spell from within the field? That is, does the spell automatically fail, and is it wasted?
No. An antimagic field suppresses magic used within, brought into, or cast into its area, but it does not dispel it (see the second paragraph of the spell description).
Does an antimagic field suppress a spell effect at the point of origin or the area of effect? For instance, if I cast silence on myself and then cast antimagic field (assuming I have the Silent Spell feat), would the entire silence spell be suppressed (because it’s centered on me), or just the 10-foot radius surrounding me, leaving a 5-foot ring of silence around the outside of the antimagic field?
It depends on the spell. Silence is an emanation, so the whole effect is suppressed if the point from which the effect emanates is inside the antimagic field. With an area spell, only the part of the area that is within the antimagic field is suppressed. Note that the antimagic field blocks line of effect, so an area cannot extend through the antimagic field. For example, if you brought the antimagic field into a fog cloud effect, only the portion of the fog cloud that is within the antimagic field would be suppressed. On the other hand, if you cast antimagic field in a corridor 10 feet wide, the fog cloud effect could not spread past the antimagic field. Note that a spread effect will flow around an antimagic field if it has space to do so, just as it would spread around any other barrier.
Does antimagic field suppress effects such as doors sealed with arcane lock, magic traps, magic secret doors, and so on?
The arcane lock spell specifically states that only destruction of the door, knock, or dispel magic can get through a door sealed with arcane lock. Antimagic field is an emanation. As such, it does not extend through or into solid barriers, such as closed doors. In your example, the antimagic field stops at the door with the arcane lock. Because the antimagic field touches the door, the arcane lock effect on the door is suppressed. (If a creature or object touches the antimagic field it is “in,” the antimagic field and any magic operating on it are suppressed.) Note that if the door in question is not a solid barrier, the antimagic field could extend through it. For example, if the door has an opening in it that is 1 foot square (see Line of Effect in Chapter 10 of the Player’s Handbook), the door does not block the antimagic field. Any magical effect operating on the door is suppressed by the antimagic field, and the antimagic field reaches beyond the door according to the normal rules for line of effect. Likewise, if the door is not a physical barrier at all, but it is a magical effect itself (perhaps a figment such as minor image), the antimagic field suppresses the magic and the “door” disappears. Most magic traps are also rendered inert by an antimagic field.
What happens when a prismatic wall, prismatic sphere, or wall of force is cast so that it intersects an antimagic field? Does the intersection count as the space being occupied, as noted in the descriptions for these spells?
Yes. These are cases in which the pre-existing spell has squatter’s rights over the space. An antimagic field blocks line of effect for any spell area or spell effect that extends into it. If you try to cast any wall effects into or through a space containing an antimagic field, the effect is disrupted just as it would be if cast into a space containing an object or creature. All three of the effects you mention, however, are impervious to antimagic field. If one of these effects is already in place, the antimagic field stops where it meets them, just as it stops where it meets a solid barrier.