Are there periodic functions without a smallest period?
For a nontrivial example, consider the Dirichlet function, which has $$\delta(x) = \begin{cases}0 & \text{ if $x$ is rational}\\1 & \text{ if $x$ is irrational}\end{cases}$$
Then $\delta(x)$ is periodic with period $r$ for every rational number $r$.
Yes, for example constant function.
In fact, a continuous function of a real variable having arbitrarily small periods is necessarily a constant. Indeed, the set of periods is then a dense additive subgroup of the real line, and the function is constantly equal to its value at any point.