The power of illusions depends a lot on the GM
So we're reading the same thing, here's the spell:
School illusion (figment); >Level bard 1, sorcerer/wizard 1
Casting Time 1 standard action
Components V, S, F (a bit of fleece)
Range long (400 ft. + 40 ft./level)
Effect visual figment that cannot extend beyond four 10-ft. cubes + one 10-ft. cube/level (S)
Saving Throw Will disbelief (if interacted with); Spell Resistance no
This spell creates the visual illusion of an object, creature, or force, as visualized by you. The illusion does not create sound, smell, texture, or temperature. You can move the image within the limits of the size of the effect.
Then you ask...
I stick an illusion of a bucket on 5 goblin's [sic] heads...
While the spell can create a figment of one object, creature, or force within the area's limits, the DM must determine if that one figment can be multiple connected things within the area's limits. Creating a figment of one (probably amusingly large) bucket on the heads of five goblins is totally legit. However, creating a figment depicting five buckets connected by ropes as one image, one figment bucket each covering one goblin head, is probably pushing the spell's limits. Then...
[...I] pass off concentration to the familiar (through some feat) [like a variant of this feat, for example] who moves [the figment] along with [the goblins].
While the area of the spell is shapeable, the area isn't mobile, despite the illusionist's ability to move the figment within the spell's area. Decisions about a spell are made when its cast unless the spell specifically allows otherwise, and that includes the area affected by the spell silent image. Thus familiar and figment could accompany the goblins to the limit of the spell's area, but the figment can't go farther. Shape the area wisely when the spell's cast.
In the case of a lone figment bucket (more of a tub, I guess), this is easily adjudicated; were the goblins to scatter within the area, the illusionist merely picks where he wants the big ol' figment bucket to go. If the GM allows the creation of multiple figment buckets connected by ropes, then things get complicated.
Tip: Decide early the versatility of illusions
The DM must decide how the spell silent image works, probably picking from among the following, but with the understanding that there are likely as many variations on these as there GMs. In increasing tiers of versatility, the spell silent image can create...
- Static figments only (i.e. once created the figment is unalterable, the figment itself unmoving). The illusionist can move the figment throughout the spell's area, but, for example, a figment of a campfire would look as though it were affected by the spell time stop, and a figment of a dragon would be stationary, not even seeming to respire, much less using its breath weapon. This is extremely conservative.
- Semi-dynamic figments (i.e. the illusionist creates a figment that engages in an orderly sequence of events). For example, in addition to moving the figment about the spell's area, a figment of a dragon could be made to appear to attack on the first round, glare threateningly on the second, and use its breath weapon on the third, but the figment engages in those actions in that order. Alternatively, the illusionist could stop the sequence, but the sequence resumes where it stopped when started again. This is moderately conservative.
- Dynamic figments (i.e. once created the figment can be altered as the illusionist desires). For example, a figment of a dragon could be altered to reflect damage that foes would've dealt it, expel a breath weapon, or fly away (perhaps using forced perspective to travel into the distance, seemingly beyond the spell's area). This is moderately liberal.
(I have played in campaigns using each of these rules, and all three work to varying degrees, but when I'm GMing I stick to option 2 as a compromise.)
Any option hinges on the phrase you can move the image; the GM needs to decide if that means an illusionist can slide the figment around, or do that and, for example, make a figment creature appear to be breathing, or do both and, for example, make a figment creature perform intricate, non-repeating dance steps or whatever. (Note, for example, the variety of opinions held in this 2010 thread.)
Going forward means picking an option or concocting your own.
Using options 1 and 2, the illusionist moves all the figment buckets connected by ropes at once as a single unit, leaving any figment buckets not on a goblin head floating in space, likely earning viewers Willpower saving throws against the spell unless the setting's overwhelmed by haunted buckets. Using option 3, however, the illusionist can keep each figment bucket on each goblin head without issue.
Then there's more! You continue, saying
Assuming they try to touch the bucket, their hands will go through. Is this treated as proof that an illusion isn't real and they can see through the bucket (even though they don't know it is a bucket)?
This will depend on the campaign. A real bucket would still allow a goblin to see sort of, and, because goblins, a goblin might even know that what's on its head is supposed to be a bucket. Assuming the goblins aren't penalized for their cranial embucketing, they just look funny—funnier—, and get saving throws against the silent image spell if (no, not when) they try to remove their figment buckets, and probably—depending on the ubiquity of magic buckets in the campaign—have proof of an illusion when their hands go through their figment buckets.
However, if the illusionist is attempting to blind the goblins with the figment buckets, that's the illusionist forcing the goblins to interact with the figment buckets immediately, and the goblins make saving throws against the silent image spell. Again, proof of the figment buckets being figment buckets should typically take actions on the goblins' parts but may depend on the setting. Further, one lucky goblin may alert the unlucky ones to the charade, granting the unlucky saving throws with a +4 bonus.
Much the same case applies to...
What if the illusion is a brilliant blazing sun directly in front of their faces? [H]ow about a blob of pure darkness?
Assuming these effects are simply dispensing with the figment bucket pretense and just blinding the darn goblins directly, these effects, too, mandate the goblins to interact immediately with the illusionist's figment. The goblins make saving throws straight away.
Bear in mind that an illusionist is limited in the figments he can create. According to the section School on Illusions on Figment
[Y]ou cannot make a visual copy of something unless you know what it looks like (or copy another sense exactly unless you have experienced it).
So while it can be assumed most illusionists have seen a bucket, it might take a level or two before an illusionist has seen a blazing sun directly before him (his own sun likely not qualifying) or a blob of pure darkness. It's probably a good idea to determine before play what kinds of things are reasonable for the illusionist to have seen well enough to create three-dimensional images of, and if to create a figment of something an illusionist must witness it with his own eyes under stressful conditions (i.e. adventuring), and if a figment can be created after the illusionist sees another's depiction rather than the real thing.
Also, remember that the spell silent image is limited to creating one figment of one creature, object, or force. But, while the buckets-connected-by-rope thing is really shady, I don't think an illusionist should be limited exclusively to figments of naked creatures.
Both because the columns are for Dungeons and Dragons 3.5 and because the column's unusual status, I hesitate to recommend Skip Williams' Rules of the Game columns "All About Illusions" (part 1, part 2, part 3, and part 4), but, if read with a jaundiced eye, the columns remain, at least, interesting.