Do support specialists get all of the same regimental bonuses as the rest of the regiment (characteristic bonuses, skills, and so on?)
Support Specialties and Regimental Traits
Yes, even when it doesn't make sense. Ogryn assigned to a Highborn Regiment creates some absurdity as do many others. You could technically use the combined regiment rules from Hammer of the Emperor and have SupSpecs from their own 'regiments' if you want added complexity.
Do support specialists get all of the same regimental bonuses as the rest of the regiment (characteristic bonuses, skills, and so on?)
A partial answer (I was looking for the same answer myself).
The Core Book page 58 says
HOMEWORLDS AND SUPPORT SPECIALISTSMany of the Specialities from Chapter III: Character Creation are Support Specialists, auxiliary characters who are attached to the Regiment to provide their special skill-set. These characters can be from worlds half the galaxy away from the Regiment’s Home World, or even have no Home World at all. Despite these vast differences, support troops often begin to adapt the mannerisms, skills, and habits of the Guardsmen they are serving with. Because this, in ONLY WAR allSupport Specialist characters gain the benefits and abilities of the Home World of the Regiment they are serving alongside, even if they were not born there.
But there is nothing precised about the Regimental doctrines so I don't know if they apply either.
The answer seems to be "to a certain degree".
It all boils down to some of the background (fluff) factors and the length of service with the regiment in question really;
To give a fluff example, Ciaphas Cain,
HERO OF THE IMPERIUM!!!
provides a decent example to personnel that is attached long after (original) regiment creation.
He has been around Valhallan regiments for most of his regular career; he prefers tanna over recaf, he is somewhat accustomed to cold (not that he enjoys it), he is fluent in the Valhallan patois of the Low Gothic in addition to the common low gothic (in fact Amberly routinely remarks on the Valhallan colloquialisms), and he has learned how to act as a vehicle crewman (to an extent, ie pintle gunner / turret operator), thou the last one can be considered either way.
On the other hand, despite all of his time with the ice worlders, he hasn't really picked up their hatred and in depth knowledge of them other than a few bits and pieces here and then, he still keeps his hive world sense of direction, he still keeps his aptitude for close combat and so on.
This leads me to think that yes, the GM should be giving some limited fitting in bonuses without going over board; give the non-native attachments a few, lighter variants of the basic, main characteristics of their regiment as a bit of acclimation bonus based on time served alongside the regiment in question.
For example, for any and every auxiliary personnel attached to a Valhallan unit, (btw I consider Valhallans to have "Ice Worlder" (ie, resistance:cold) talent, unlike the HotE) I would add Cold Acclimation (0-5 bonus to checks involving cold. Depends on the length of service) talent and possibly a few other small bits if the situation warrants it and leave it at that.
As for Daze's specific example, I would generate the Ogryn with using his own regiment (Ogryn world, line or close assault or grenadier, same commander as the normal regiment with doctrines and equipment choices being about half/half from both regiments to emphasis the nonstandard gear and training) which negates issues like +20 tech use of void borne, grenadier, sapper Ogryns
In the specific case of Ogryns, no I wouldn't give him linguistics but I would give him the capability to speak Ogrynised High Gothic along with his Low assuming he has been around highborn types for a decent amount of time.