So what are the attributes of this molecule we have just designed? Was it worth it? Judge for yourself:

Some of the attributes below are guesses, and may be way off.

Any size from a few nanometers up to extremely large e.g. a skyscraper.

Very low weight (and mass).  Since this material has lots of holes in it, it weighs very little.   {See mass estimates for more details}

Hopefully very high. I do not know exactly how high, but it will be not as strong as solid diamond of the same size, but much, much stronger than solid diamond of the same weight (mass). (My guess is it will be stronger than steel, but perhaps it will just be as strong as Jell-O.)

Unknown. Probably clear like diamond, especially considering its gas-like density. Possibly white like snow due to all the scattering going on in the myriad internal surfaces. Possibly cloudy or milky like quartz crystal. Appearance would depend heavily on configuration.

Ductile and not at all brittle. Can be pre-stressed like spring steel. In fact, it may have up to about 25% spring in it, so will probably need to be pre-stressed in applications that need more rigidity. Another way to put this is that is has high toughness and energy absorption capability (high energy-to-break).

Unknown. As suggested above (in Strength), probably much higher than diamond. Larger scale objects will have a higher strength to weight ratio (I think), due to there being more and larger 'holes'.

Highly variable. See configurations.

Carbon atoms in this formation can withstand a large temperature range (probably similar to nanotubes, which burn at 900 degrees Celsius).

Unknown, but possibly highly reactive - may need to be enclosed in some way to prevent reactions (and moisture build-up).

How long would this material last? Considering all its other attributes, the answer is: a very long time, on the scale of decades, assuming it was protected from averse reactions.

Yes. Since it is made entirely of carbon atoms, it would be relatively easy to reuse the atoms and turn it into something else. You would not even need to break apart all the atoms - you could reuse the structures at many of the levels. (Of course this does not take into account the energy cost of doing so, and whether it would be economically feasible.)

Carbon nanotubes of the zigzag type like those in Buckymesh conduct electricity and can reach a very high current density with very low resistivity. (See Equilibrium Structure here for an explanation of zigzag vs. armchair. See Electrical Transport for conductivity details.)  I don’t know what the current density and resistivity would be of carbon nanotubes in this formation. Due to the heptagons, there may be significant phonon scattering at each of the junctions, causing low electrical conductivity.

Very high. Also, under very specific conditions carbon nanotubes are flammable, and hence the design may need to be modified to prevent this. For example, the top level structure could perhaps be encapsulated in some way (which would have other useful side-effects, such as preventing a build up of moisture).

If a low-level strut fails (e.g. is physically broken), the effect on the whole structure is unknown at this stage. However it has been suggested by Richard Smalley and appears also from other research that breaks in nanotubes may be 'self-healing', and hence a failure of a small subset of the struts may not significantly reduce the effectiveness of the structure. (Of course, he may be wrong, or nanotubes in this configuration may not behave like isolated nanotubes, and the whole thing may unravel.)

I do not have the tools to calculate the exact values of strength and elasticity rigidity (e.g. Young's modulus and tensile strength) for different scales - this is left as an exercise for a clever reader. Ditto for several of the other measures.

Many of the characteristics of this material stem from the characteristics of nanotubes. See also here.