@Nuff_SaidThis is a complex one, especially the start.
For the GGF, realize the "Russian Revolution" was over 6 years long (officially), with many placing it more at around a decade. Many factions, and sides changing all the time. White Russians, Bolsheviks, Mensheviks, Anarchists, and ultimately the Communists taking over and eliminating most others (including the Bolsheviks - they were not really the winners).
So when that soldier returned home, where, why, who did they support? Did they flee because the almost decade of war and anarchy was more than they wanted, or because the side they favored (and maybe supported) was destroyed?
GGF service is simple, forget engineer and mechanic, make him Infantry, and an Unter-offizier. There were no real "mechanics" or "engineers" in the Russian Army, you were Infantry, Cavalry, or Artillery.
Oh, and he would have to decide to bail out very early. The White Army was quickly forced to retreat East, into Siberia. That is why many ended up in the fringes of Japanese occupied regions on China and throughout the Far East. The White Army was pretty much out of Europe by the end of 1918. Otherwise they would have not been able to easily emigrate to the US, being stuck in a place like Shanghai as a "stateless person". Their old country no longer existing, they new one not recognizing them. Instead, consider having them as coming from a border region, like say the Ukraine, and after some time in a German POW camp deciding upon release to remain in Germany or Poland, then emigrating.
Service for such in a WWII Army, not likely. Unless he had gained some particularly valuable skills, like medical or logistics. Say if he had run a freight company. Most of those recalled for WWII were really specialists and experts in something the war effort really needed.
For the GF, that all depends on what he did. In the Army, that would have seen him thrown into Normandy in 1944, then fighting his way across Europe. Unless he was sent to the Pacific, where he would have been in battles like Guadalcanal and Okinawa.
The father, would be reaching draft age at around 1966. Now contrary to popular belief, not many draftees served in Vietnam. If you were drafted, you served for 2 years. And were normally kept in the US, or sent to Europe as part of the still ongoing occupation duty. Trained as say a mechanic, and working on tanks for a year in Germany before going home.
But "draft dodger"? Largely a joke. Just deciding to go to college in that era made you exempt from being drafted until you graduated. So graduates school in 1965, stays in college until 1970, by then simply be married and have a kid. That classified you to where being drafted was almost impossible as there were plenty of single younger men they could grab.
You need to decide a bit more how you want to approach this, then I could give some better ideas of how to pull it off. But realize, many of the terms you are trying to use at turn of the century times are not quite what you think.
"Engineers" at that time were working on things like canals and planning rail lines. That made them useful in laying out things like trench lines, drainage, and laying roads and rails for logistics. They are not "engineers" as most would think of them today. And mechanized equipment was so new, "mechanics" were nothing like what is seen either. A train engineer drafted, then spent their time running trains during the war.
By WWII, especially in the US Army, there it largely depends on the theater. Most went to Europe, some went to the Pacific. If Infantry, just watch "Band of Brothers", and forget the air drops.
For the father, absolutely no idea. He would not fit the "draft dodging hippie" segment, that was more 1968 and after, when the antagonism against "The War" was reaching it's peak. And even then, the issue was not that they were thought of as "cowards", but simply "not patriotic". That was a counter-culture thing, refusal to follow authority more than anything else.
For the MC, once again, almost impossible to tell as there are so many questions open. But by the time he is "aware" of the real world, it would be 1978 or so, the era of Disco. Carter was President, all "Draft Dodgers" were given an amnesty, and the Vietnam War was all over. The father would more likely see issues with classmates that did go and fight than from his father and grandfather.