The wizard travels to a false destination
That is, if the wizard's once-familiar destination now doesn't exist or has become unrecognizable, the wizard is, perhaps, dealt some damage and either arrives at a similar destination or the spell simply fails. As per the spell teleport:
"False destination" is a place that does not truly exist or if you are teleporting to an otherwise familiar location that no longer exists as such or has been so completely altered as to no longer be familiar to you. When traveling to a false destination, roll 1d20+80 to obtain results on the table, rather than rolling d%, since there is no real destination for you to hope to arrive at or even be off target from.
Emphasis mine. The spell greater teleport (which was teleport without error in Dungeons and Dragons, Third Edition) doesn't eliminate the chance to accidentally teleport to a false destination, eliminating instead the chance of randomly being off-target.
A false destination means, instead of rolling on the usual teleport chart, the DM rolls 80+1d20.
On 81-92 the wizard is in a similar area:
You wind up in an area that’s visually or thematically similar to the target area. Generally, you appear in the closest similar place within range. If no such area exists within the spell’s range, the spell simply fails instead.
On a 93-00 the wizard experiences a mishap:
You and anyone else teleporting with you have gotten "scrambled." You each take 1d10 points of damage, and you reroll on the chart to see where you wind up. For these rerolls, roll 1d20+80. Each time "Mishap" comes up, the characters take more damage and must reroll.
Thus a wizard attempting such travel to his destroyed laboratory either arrives at a thematically similar location (e.g. a brewery, an alchemist's shop, or another wizard's laboratory—surprise!) or, if no such thematically similar location exists within the spell's range, remains where the spell was cast. In addition, with enough bad luck, the wizard might also be dead.
The image can't cast dimension door.
One of the lines of project image that you've listed says "The projected image can't cast any spells on itself except for illusion spells." Dimension door has a Target of "you and touched objects or other touched willing creatures". Ergo, since dimension door has a Target of "you", and isn't an illusion spell, Alice can't cast it through her image.
Since dimension door always teleports the caster as well as possibly teleporting another target, the correct way to interpret the admittedly-ambiguous phrasing "you and touched objects or other touched willing creatures" is "You. Additionally, touched objects or creatures." There is no way to cast dimension door in a way that doesn't affect "you", so it will always have a target of "you", and thus be un-castable by a projected image.
Best Answer
The spell fails and is wasted.
According to page 171 of the Player's Handbook, "If you ever try to cast a spell in conditions where the characteristics of the spell (range, area, or the like) cannot be made to conform, the casting fails and the spell is wasted."
The definition of "spell characteristics" given in that quote is the only one we have, but it seems reasonable to assume that the maximum distance the teleport spell can achieve is one such spell characteristic. It is, after all, a characteristic of the spell.