I am going to do the first two in reverse order:
> **Puzzle 98.** What does a category with exactly one object amount to? It's something we've already discussed!
These are [monoids](https://en.wikipedia.org/wiki/Monoid).
> **Puzzle 97.** How many different categories are there with exactly 4 morphisms?
This question involves a lot of counting.
Here's a lower bound: There are at least as many categories as there are monoids of order 4. According to [OEIS A058129](https://oeis.org/A058129), there are 35 monoids of order 4.
There are just the categories with one object. There are also categories with two objects, three objects, and one with four.