In 2003 IDE Secretary, Phil Stroud, received an enquiry about the 'strange' horns in the IDE logo and a request to know what they meant. The enquirer was also mystified by the 'strange blobs' in the monochrome version of the logo.
I believe that the logo represents the earliest recorded demolition project.
The actual date when it took place is only a very rough estimate. The full story can be found in the Old Testament, The Book of Joshua, Chapter 6.
The project, as far as one can gather, was to flatten the walls of the city of Jericho. The method statement for the demolition involved sound vibrations and the contractor was called Joshua. Basically the city walls were encircled by priests blowing seven rams horns, seven times over a period of seven days. On the seventh day all of Joshua's men were instructed to shout all at once. This apparently did the trick. The walls just fell down and Joshua and his men were then able to enter the city, finish off the opposition, and no doubt set about the task of recovering non-ferrous metals (gold, silver, etc.).
I have not heard of anyone attempting this particular demolition technique since that time. I think the method would, in view of the volume and intensity of sound needed, have quite serious health and safety and environmental implications. The time-scale of a week to demolish an entire city wall is not bad going though!
So that explains the seven horns in your question. As for the "blobs", I am quite happy to accept your descriptions of "phoenix" and "spiders". However looking at an enlarged version of the logo I think these are meant to represent palm trees. Why palm trees? Presumably Jericho had palm trees.