Describes doing something with no trouble, preparation, or fuss; often implies natural ease rather than laziness.
He solved the difficult problem with ease.
She just casually tied up her hair, but it looks very stylish.
Often carries a nuance of doing something without much thought, care, or formality; can be positive (relaxed) or negative (sloppy) depending on context.
He casually placed the documents on the desk.
I gave an offhand reply and later regretted being rude.
He carelessly set his bag down on the table.
She casually tied her hair back and went out.
He pulled off that difficult trick with total ease.
適当 suggests doing something in a random or perfunctory way, while 無造作 emphasizes doing it naturally and without conscious effort, which need not sound negative.
無造作 describes an unaffected, effortless casualness that can even sound positive, while ぞんざい almost always carries a negative, careless-or-rude nuance.