Tempera Grassa & Sun Thickened Linseed oil

Hi Maggie,
Welcome to MITRA.  For illustration work, good at details and fine lines, no fumes, the mediums that come to mind are:
1.  Gouache.  Fast drying, water-based binder (gum Arabic).  As a non-polymerizing paint (remains soluble), gouache can’t be indefinitely layered (at some point fresh layers dissolve underlying ones) which limits glazing.  Because gouache doesn’t cure it may not meet your idea of “archival” (if by archival you mean an insoluble paint that holds up well over time. Technically speaking, the word ‘archival’ refers only to works on paper.  For paintings, I believe terms such as ‘long-lasting’ or ‘durable’ are used.) 
2.  Acrylic.  Fast drying, water-based binder (synthetic polymer) that cures (becomes insoluble) – hence can be developed in many layers, with endless opportunities to glaze.  
These two options are simpler than Tempera Grassa (TG) because…
–  Support & ground requirements for gouache/acrylic are simpler (TG needs more rigidity & absorbency in support/ground) 
–  Their handling properties are a bit easier to learn than TG (which, as with pure egg tempera, is prone to lifting until one gets the hang of it).
–  There are many good quality, commercially produced, tubed gouache and acrylic options available (versus TG, which is generally made from scratch. I know of only one commercial TG option, Sennelier tubed “egg tempera” paints, actually TG; see http://www.sennelier-colors.com/en/Gouache-tempera_84.html)  
Henceforth I’ll mention both egg tempera (ET) and tempera grass (TG), since they overlap.  The major difference is that TG has a bit more flexibility (because of oil content) and is a bit more painterly. 
Given ET’s and TG’s challenges (difficult to handle until one gets a feeling for the paint) and inconvenience (made from scratch), most people don’t willingly choose these mediums.  Instead, people fall in love with their attributes (fast drying, endless layering, exquisite linearity, great faux effects, etc.) and thus are willing to “put up with” the challenges and inconveniences of ET and TG.   This isn’t to deter you, just to clarify what’s been my experience (having taught these mediums for many years).  
But, you may have reasons for preferring TG over gouache or acrylic – so I’ll address your specific questions, best I can.
 1.  Could I use an egg oil emulsion as a quick drying medium?  Pure ET paint (yolk + pigment, no drying oil added) is fastest drying.  Combining oil + yolk to make an emulsion (TG) makes it dry a bit more slowly, but still within seconds for a thin stroke of paint.  
As long as there is more yolk than oil in the TG mix, the paint is water-based and dries fairly quickly.  More oil than yolk in the TG mix makes it solvent-based, with more of the handling properties and drying time of oil paint.
Although ET and TG paint are touch dry in seconds, the paint doesn’t “cure” (become insoluble in its diluent, be it water or solvent) for weeks.  So while you can layer ET and TG paint indefinitely, even before the paint has fully cured, it takes a light touch and practiced hand not to lift underlying layers which, until they are fully cured, remain vulnerable to lifting as additional layers are built on top.   The challenging learning curve of ET and TG is how to build layers without lifting paint.  
2. I’m painting on acrylic gesso.  The ground for ET and TG is complicated; I don’t want to overwhelm with info but don’t want to misinform you either.  I’ll try to keep it simple….
ET and TG require two things in a ground:
1. Extra Absorbency
2. Rigidity
Extra absorbency is needed to (a) absorb water content in the paint, which helps control the paint and minimizes lifting; and (b) create good adhesion – the paint seeps into and interlocks with a porous, absorbent ground. 
Most acrylic gessos are not absorbent enough.  However there are a few synthetic polymer gessos specifically created with more absorbency.  They work okay for ET and moderately well for TG.  For ground suggestions, see this article:   https://www.naturalpigments.eu/artist-materials/comparison-tempera-grounds
Rigidity is important because, as you note, ET and TG aren’t very flexible paints to start, because…
a. They have a high pigment content (binders bend to a degree but pigments don’t bend).
b. Egg yolk binder becomes more brittle with age (because plasticizing egg oil moves around and can exit the paint film, decreasing flexibility).

You can paint a few thin layers of ET and TG on flexible supports, but if you build up any substantial amount of either paint you should use a rigid support. 
Would it be fine to use [acrylic gesso or TG?] as long as I: Dont use it in the underpainting  Dont exceed 25% medium by volume  Or is it only suitable for glazing?
Not sure what you mean by the above…are you planning to use acrylic gesso in the painting itself?  Not a good idea to combine acrylic gesso/paint with ET or TG in any way.
 3.  Could I use stand oil instead of regular linseed oil to make up for the brittleness of the egg? The brittleness of ET and TG comes in part from the yolk binder – but equally so from the high pigment content in ET and TG.  As mentioned, pigments don’t flex, so paints with high pigment load are less flexible.  No matter the binder, you can’t get around the high pigment load. The oil content in TG does make it a bit more flexible than pure ET, but still it’s not as flexible as pure oil or acrylic paint.    
I don’t believe stand oil is more flexible than regular linseed oil.  Stand oil is faster drying than regular linseed because, like sun-thickened linseed, stand oil has been partially cured/already polymerized (by heat in the former, sun in the latter).   But I’m not an oil expert – others can chime in.  
4. I’m also using safflower oil to clean my brushes, would egg residue contaminating the oil cause problems or spoil? 
As long as you don’t swirl the brush in the safflower oil, so that paint gets into the safflower, it shouldn’t be a problem.  Just dip the brush tip into the oil, wipe it on a towel to remove paint, and keep doing so until most of the paint is out of the brush. Then wash the brush in water and soap (I use a pure castille bar soap) until the water runs clear.   
5. Sun Thickened Linseed Oil 
Sun-thickened linseed is faster drying than ordinary linseed, and an experienced painter can achieve fine line work and detail with it – but in general I don’t think of oil paint, even sun thickened, as an ideal medium for illustration work.  
6.   Is Tad Spurgeon’s website and book considered a reliable source?  I actually stumbled onto Louis Valezquez while researching egg oil emulsions a couple months ago.  I own and enjoy Tad’s book but I’m not primarily an oil painter, it’s best to let the oil experts comment.  I will note that I don’t believe Tad is primarily an ET or TG painter, he’s an oil painter and that’s where his expertise lies.  I’ve corresponded with Louis Velaquez and have his book – his enthusiasm is admirable but I would take his thoughts on ET and TG with a grain (or more) of salt. 
Hope that helps.  Koo Schadler