I don't know many specifics of Korean grammar but I will offer a few comments according to my limited understanding, beyond that, I am not qualified to explain tense and other complex or rare grammatical exceptions:
-더니 is a connective ending, am I right? What is the difference between that and -다가?
~더니 - is conjunctive between two clauses that expresses the action of the first clause is a reason, in other words 'because of this' or 'since' then the following occurred or is true.
(나는) 그 일을 직접 하더니 얼마나 힘든 일인지 깨달았어요 - I realised what a difficult job it was since I did (was doing) the work myself.
그 일을 직접 하시더니 얼마나 힘든 일인지 알게 되셨나요? - Since you had a hand in doing the work did you come to understand how difficult it is?
얘는 빈 깡통을 보더니 발로 찼어요 - Upon seeing an empty can the child kicked it. Since he saw the empty can he kicked it.
~다가 - is also conjunctive between clauses but rather indicates that there was a discontinuance of the action of the first clause and a second (yet related) action or result occurred, this may also indicate a change of location between where the first and second actions take place (this pattern should not to be confused with other patterns that have 다가 as an element, such as ~은/는데다가 ).
그 일을 하다가 쉬었어요 - I worked for a while then rested. After doing the work I rested.
요리를 하다가 먹었어요 - We cooked the food (in the kitchen) and then ate it.
한참을 벼르다가 철수는 드디어 하고 싶은 말을 꺼냈어요 - After looking for just the right opportunity, Chul-soo finally brought up what was on his mind (what he wanted to say).
There's also some kind of rule on the speaker of each clause? Like -더니 you can't use in the past tense unless you're talking about yourself etc etc., am I right?
There are always grammatical rules and exceptions but who has time to think about them when having a conversation? I acknowledge, however, their importance when it comes to writing and translation. Some strange things happen when considering tense in these patterns, for example a true 'past tense' (e.g. 했더니, 했다가) does not seem impossible but I think it would be a rather rare construction in everyday speech. And while a future tense appears to be impossible in either case, when we apply future tense to '더니' something unexpected happens 그 일을 하겠더니 기뻐요 -Since you are saying (now) that you are going to do the work (in the future) I am happy.
Can you please provide examples too!
Yeah well maybe some kind volunteer will come along and give some great examples and correct all my mistakes -- but I wouldn't hold your breath while waiting.
George Posten - fellow student 한국명은 나영훈입니다