Assume Smoldering Egg has no damage marked on it at step 1.
Then by step 6, Ashmouth Dragon will be 4/4 with 2 damage marked on it (not 4/2 — in general, damage dealt to a creature doesn't reduce toughness [C.R. 120.3e]; compare Crush the Weak with Necrotic Wound). Here, the damage marked on Smoldering Egg carries over to Ashmouth Dragon since a permanent that transforms is the same object as before (C.R. 712.14).
When the second Crush the Weak resolves, it will deal 2 damage to Ashmouth Dragon, for a total of 4 damage marked on Ashmouth Dragon. This is lethal damage, so Ashmouth Dragon is destroyed as a state-based action (and is exiled rather than dying) (C.R. 117.3b, 117.5, 704.5g, 700.4, 701.7a, 120.6).
2. Crush the Weak is played (full cost and by the same player who owns Smoldering Egg)
3. Smoldering Egg is now 0/2 with 6 counters on it
4. Crush the Weak is played again
5. The counters go on the stack and Smoldering Egg is transformed to Ashmouth Dragon
6. Is Ashmouth Dragon at 4/4 or 4/2 due to damage at step 3? When the second Crush the Weak resolves, will it kill Ashmouth Dragon?
Then by step 6, Ashmouth Dragon will be 4/4 with 2 damage marked on it (not 4/2 — in general, damage dealt to a creature doesn't reduce toughness [C.R. 120.3e]; compare Crush the Weak with Necrotic Wound). Here, the damage marked on Smoldering Egg carries over to Ashmouth Dragon since a permanent that transforms is the same object as before (C.R. 712.14).
When the second Crush the Weak resolves, it will deal 2 damage to Ashmouth Dragon, for a total of 4 damage marked on Ashmouth Dragon. This is lethal damage, so Ashmouth Dragon is destroyed as a state-based action (and is exiled rather than dying) (C.R. 117.3b, 117.5, 704.5g, 700.4, 701.7a, 120.6).