Answer for current main problem
You may assume that the mother will search for the soldier if you want.
If you are assuming that both parent care for the offspring, she can wait for the male (or call him) and leave him with the other eggs. Realistically, If the egg was taken by a predatory animal, the egg would have been eaten already, but if the dragons killed every predator that destroys their eggs, the predators would learn to avoid dragon nests, so revenge may be a justifiable strategy for dragons (and if their instinct is not too inelastic, the mother will protect her eggs when she'll see that they are whole by now).
If dragons had much to do with humans, they could have instinct that tells them to pursue a man that has taken their egg. It seems to be realistic that the man will understand that the dragon wants the egg back.
Even solitary dragon can hunt egg-thief if most predators are already afraid of dragon revenge. On the other hand, this strategy may also not be used if you assume that most eggs-thieves will not learn or that the dragons are to primitive to use such strategy even by instinct. Similarly, even if dragons have much to do with humans, their instinct may make them do something else like killing the human or escaping (with or without other eggs). Once again, you may assume what you want.
If dragons are intelligent, the problems changes. (But I'm thinking that you have in mind an idealistic view of dragons as animals.) Now the dragons should understand that killing humans is immoral and humans should understand that killing adult and unhatched dragons is immoral, so they should find a peaceful solution. This may be easy if there is no deep-rooted hostility.
Answer for original question
You may assume anything.
Dragons may be like cats and crocodiles (females care for the young), like three-spined stickleback (males care), the pair may hunt and guard nest in turns (that would also justify one dragon attacking a city without its mate) or they may guard nest together for the most time (and attack together when they recognize a city as a threat).
They may lay eggs in the spring, in the winter (for example because then they will hatch in the spring, but maybe hunting in the winter is easier), in the autumn or in the summer. The eggs may hatch immediately (ovoviviparity), in one day (rather for some magical reasons), in weeks, months, years or thousands of years (magic would make it easier to justify again).
Cold may make dragon fire weaker or not and it may even make it stronger (for example if better cooling of one part allows for making another part hotter).