Are there any proofs of exponential lower bound time complexity
The Time Hierarchy Theorem states that for any (reasonable) function $f$, there exists a problem that cannot be solved in time substantially faster than $f(n)$. The proof of this is similar to the proof of the undecidability of the Halting problem: we construct the decision problem "given a description of a Turing machine, does it halt in time at most $f(n)$ and then show that it cannot be decided substantially faster than $f(n)$, in a very similar way to showing that "does this Turing machine halt in finite time" cannot be decided (in finite time).

In the case of generalized board games (e.g., Chess), these are often $EXPTIME$-complete. This means any problem in $EXPTIME$ can be reduced in polynomial time to it. Since (by the Time Hierarchy Theorem) there are problems in $EXPTIME$ that cannot be solved in subexponential time, neither can Chess (or any other $EXPTIME$-complete problem).

Your other example (shuffling bits to get a prime number) probably doesn't require exponential time. The chance that a random permutation of the bits results in a prime is quite high, and only a polynomial number of guesses would be required to find a prime. The problem is in $NP$, so a proof that it requires exponential time would be a somewhat significant breakthrough.