How do we pick and switch hash functions from a universal family?
You need to decide what sort of resilience you are expecting from your hash function. While it is true that once picked, your hash function does not change (and in a sense &quot;the rest of the family doesn't exist&quot;) but you're ignoring the crucial part of picking the function at random.
Suppose that $\mathcal{H}\subseteq U^{[m]}$ is a universal family, $h$ is chosen uniformly at random from $\mathcal{H}$, and is subsequently used to hash elements from $U$ into $m$ cells via chaining. This guarantees that regardless of what is the current content of the table, if the load factor is $\alpha=n/m$, then for every $x\in U$ (note the worst case analysis, as opposed to the average case analysis in the simple uniform hashing case) the expected search/insertion time for $x$ is $O(\alpha)$, where the expectation is taken over the initial choice of $h$ (this is the only place where randomness occurs). This remains true regardless of how many operations were done since the initialization, the random choice of $h$ guarantees low expected search time for all inputs.
Switching might come to mind if you are looking for robustness against certain types of adversaries. If the entire problem was handling a user giving you inputs and hashing them to a table, then everything's fine. This changes if you are looking at the problem of an adversarial user with more access, say e.g. that he might learn the hash value of some elements (which happens in real world attacks based on response time analysis). In that case, the adversarial user might be able to recover $h$ and then send you tailored inputs that will cause high search time. This relied on the adversary having a richer interface than just being able to send arbitrary values (which alone, as we saw above, can't cause problems). Now you're entering the world of cryptographically secure hash functions, and here some switching mechanism might be of use. This requires an entirely different formalization, and to say something interesting about parameters such as switching intervals we need to be clear about what sort of robustness we want to achieve.