This is a bit of a touchy question because there is no clear answer as regards the rules, I'll provide the assured answer, and then the interpreted answer with my reasoning.
1. Can his power affect his own unit if he is embarked with them?
- Clearly yes - he has the Master of Pain rule, so Father doesn't matter)
--------------thus endeth the clear part
2. Would it affect other units within 12" of a transport he was embarked on?
- Most likely yes, the relevant rule would be on Page 80 Embarking & Disembarking first paragraph of the 'Embarking' rules the last sentence. Basically it does clarify that if you need to measure a distance to them that is not shooting, you measure from the hull...well, what would you need to measure a distance for that is not shooting? Basically bubble effects. There is no rule that either clearly allows or disallows this, so it seems reasonable to assume that his bubble effect would work. I have been at tournaments that argued against it, and can understand their thinking, but have also been at tournaments that allowed it and understand theirs. I tend to favor the latter, personally, because it makes sense.
3. Would it affect other embarked units if Urien was not embarked with them?
- Most likely yes, again, this is argued with the above rule. Functionally you can still measure 'ranges' to them and there is nothing disallowing his effect (unlike, say, a psychic or shooting effect, both of which are specifically ruled out as being able to affect passengers).
There is no evidence to suggest that Father of Pain works in any way like a psychic blessing - it is not classified as a psychic power nor a blessing.