I won't directly answer your questions but consider this. We have a function F that accepts an argument of size N and returns the result after N² instructions. Its time complexity is O(N²).

Now consider F'. F' is like F but we limit N to a fixed constant -- for example we say that N must be ≤ 2³². In this case, _for every input_ F' returns the result after at most (2³²)² = 2⁶⁴ instructions. Its time complexity is O(1). Note that O(2⁶⁴) = O(2⁶⁴ * 1) = O(1).

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Technically, all implementable-on-real-machines algorithms either return after O(1) instructions, or loop after O(1) instructions. But being O(1) doesn't imply being fast in practice.

You might be also interested in reading about galactic algorithms: https://en.wikipedia.org/wiki/Galactic_algorithm.