Beyond the textbook.
Yes, many of the oldest ones do. The character began as a small picture of the thing, and was then simplified over time.
How the shape connects to the meaning: the character keeps the outline of the real thing. Once you know the picture it came from, it is easy to remember.
The Chinese script is logographic, with characters representing entire words or morphemes (smallest meaningful units of language), rather than sounds. Explore more examples of Chinese characters that resemble the objects or ideas they represent. Note down at least two such characters and explain how their shapes are connected to their meanings.