OK so I gave the game another try after your instructions and wow, you really need to write that down in the game page because it's not clear at all in the first screen.
1) Pick up and throw dragoons and eggs using SPACE BAR
2) Eggs explode and temporarily make surrounding dragoons purple. Doki can walk through purple colored dragoons.
3) Eggs are not passable, no matter if they're exploded or not
4) Use your chain (SPACE BAR) to grab longoons in the distance to pull yourself next to them (you need to be aligned vertically or horizontally)
Knowing all that, now I can solve the puzzles!