I'll add a caveat that there are other notions of completeness and expressivity that are not well modelled by Turing machines. For instance, the pi-calculus is Turing complete but can express concurrency. It is known that more expressive process calculi exist than the pi-calculus however. Similar issues arise in higher type computability.

To have a trailing zero you need a 5 factor for every 2 factor. This will never have any 5 factors and will thus never have any trailing zeros. This problem can be solved in constant time by returning zero.

If you're a tad clever you can reduce this to a Fibonacci like problem. The result will be a power of 2. What happens if you try and calculate log2(f(n)) instead of calculating f(n)? Also zeros in what base? base 10?

To argue via the connection between them you'd have to argue that you could compute the convex hull faster than that if you had a faster algorithm to compute triangulation. Also, can you define what you mean by "getting a triangulation" others might be familiar with this term but I am not.

Not if you represent the count in binary.

What if you keep a count or two? How many bits are in a count? How efficiently can increment/decrement a count?

Another way of looking at this that depends on your philosophical position on the state of human intelligence: Humans are a physical process, Turing machines should be able to simulate any physical process, therefore a Turing machine should be able to simulate a human.

How do we know about 1/n are irreducible? I've read that in "fast construction of irreducible polynomials over finite fields" (well actually 1/2n but same difference). The reference hits a thick textbook though.