While the question is tagged as dungeons-and-dragons, my answer is tailored toward AD&D as the OP references that game as the one being played. The basic concept can be applied to any version but the analysis may well be too AD&D specific for all versions.
Justifying why monsters are working together is important to a degree. Sadly, in most cases, players don't question it unless it's an odd coupling. As example, drow and elves working together. That said, it is always a good idea for a GM to have some kind of back-story to cover groups of monsters working together. That just makes things more interesting. Don't be discouraged if the players don't actively seek out the story.
It is reasonable to have a dragon exploiting nearby monsters for its own purposes. A smart, strong monster (like a dragon) will seek every advantage. Allies are an advantage as are plebs. In this case a green dragon being worshiped by a band of troglodytes could happen. I think it more likely that the dragon is intimidating the trogs to do as it commands. Kind of a "bully of the block" scenario. Likewise, that may be the relationship between the trogs and kobolds. The reason I lean toward a bully scenario is the low intelligence of the trogs. It is likely that they don't understand enough to allow for a complex relationship but they get the concept of, "Do it or I'll eat you."
Now, if the dragon is a speaking dragon then a more complex relationship could evolve. Such a dragon, assuming it speaks their language (as GM if you want it to it does), may well act as an oracle of sorts if the wee ones bring it shiny stuff or fresh meat or other things it wants. In that case worship is more likely. Magic use on the dragon's part is good for shock and awe with the trogs but not overly necessary.
The key to any such relationship is, what does each party get out of the arrangement? A smart dragon may want scouting done or lookouts or some other task that is too menial for a great wyrm. The trogs may want help or just not to be looked at as lunch. The kobolds may want the same.
The relationship between the trogs and the kobolds is a bit more problematic. Both are not bright and it is therefore reasonable to say not prone to talking to something not of their tribe. There would need to be some reason for them to get together and form the bond they have. It could be a simple bully relationship but I think the trogs would just eat them. Maybe the kobolds fit in smaller tunnels and fetch things for the trogs from those tunnels. Maybe it is something the trogs need for the dragon, directly or indirectly. As you suggested, the dragon may well be the binding agent. Whatever the reason, it can be as simple or complex as you like but when dealing with the dumber monsters simple is better.