Can a list of hashes with property $X_{n+1} = \operatorname{SHA-256}(X_n)$ loop?
Is SHA-256 designed to loop after exactly $16^{64}$ hashed hashes, or is there a chance it will do so before that point?


SHA-256 behaves more or less like a random function. You can expect it to loop after approximately $\sqrt{2^{256}} = 2^{128}$ iterations due to the birthday paradox