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(koo schadler)
Hi Eli,
Your question is hard to answer as it's dependent on so many variables (hole size, how much water in your paint, how many paint layers on top of the holes, is the painting protected [varnish or under glass], your expectations, etc.). Here are some thoughts.
Ideally it's not a good idea to have holes of any size in gesso; they're entry points for moisture (dust, dirt too) which can be harmful to paint layers, gesso and support. But how potentially problematic depends on variables. If there are many holes, and they go all the way down to the panel, and you work with very watery paint, and the painting is kept in a moist environment, and there is no varnish or glass framing, and you have expectations of longevity/durability - well, then the pinholes are likely not a good idea. If you can minimize them (very few, keep them from going all the way down to the panel) and protect your work, then they're likely not much of an issue.
Yes, you can fill them by "spackling" them, either with fresh gesso or a commercial ground suitable for ET (such as Natural Pigments Tempera Ground. FYI, this is also how to address pinholes in gesso). Or, you can fill the holes with egg tempera paint, although this process can take a while. Neither is ideal, but likely not very problematic either.
I don't know of any comprehensive study of SID and egg tempera - excepting a single test panel I made years ago. I did a test because when I stocked up on commercially produced panels (from True Gesso) for workshops, made with chalk & animal gesso on untempered hardboard, I noticed that as the panels sat around in my studio they yellowed very slightly, in a way that suggested SID (see attached image). 
SIDS.jpeg I could imagine relative humidity traveling in & out of the panels being the culprit. So I did a test panel per Golden Artists Colors directions. The result was was very minimal discoloration on the test panel (see below). But one test panel does not make for a conclusive study.
Anecdotally, when I or students have started with slightly discolored panels I've never seen visible consequences in the artwork. But, perhaps, the many layers of paint and varnish I apply counter this. Perhaps if one were to use very watery working methods that could draw up lots of impurities in the hardboard, and paint very thinly so the underlying gesso shows through, and the hardboard remains exposed to moisture (via pinholes), with no varnish...well, perhaps the discoloration would show (or increase). I don't know. Until more testing is done it's hard to make definitive pronouncements about SID and egg tempera. Amidst uncertainty, I'll say that I haven't yet seen compelling evidence that SID is a concern for ET.
FYI: The results of my SID test were complicated & I don't want to get too off track by describing it - but you may be curious...so here, in brief is what happened. I applied three coats of GAC 100 to write the word 'SID' directly on a hardboard panel, then applied 6 layers of chalk and animal glue gesso on top. If SID occurred, the word painted with the GAC 100 would block migration of impurities, and as the panel yellowed a faint, lighter image of the word SID should appear on the panel. This happened but only very slightly (barely perceptible) and only with the letter I. The surprising result was that the gesso sunk in slightly where the GAC 100 lettering was applied underneath (see attached image; the faint image of the S and I are not SID discoloration, but shadow/light effects from indentations in the gesso). 
SIDS Test Panel.png The panel was photographed in raking light, the indentations are much less pronounced in flat light. I don't know why this happened. FYI, I also did a cross hatch adhesion test to see how chalk and glue gesso adheres to GAC 100, it seemed to adhere well, although generally I recommend against applying true gesso over synthetic polymers (see how complicated this can get?!) Again, more testing is needed...but, given that I'm not a conservator and experiments take up precious studio time and SID and egg tempera don't seem to be a big issue - well, I haven't pursued testing any further. If you do any testing, please let us know!
Happy painting,
Koo
