What is Finite Rate of Innovation Signal?
If a signal can be exactly represented by $N$ real numbers per time interval, then its number of degrees of freedom for that time interval equals $N$. The most well-known example are band-limited signals, which can be represented by $2B$ samples per second, if their bandwidth is not greater than $B$. I.e., such signals have $2B$ degrees of freedom per second. This is equivalent to saying that these signals have a finite rate of innovation, and that rate of innovation equals $2B$.

A signal doesn't need to be band-limited in order to have a finite rate of innovation (i.e., a finite number of degrees of freedom). You can think of any signal that can be parameterized with a finite number of parameters per time unit. A simple example would be a signal of the form

$$x(t)=\sum_kw_kg(t-kT)\tag{1}$$

where $g(t)$ is a rectangular impulse with a value of $1$ in the interval $[0,T]$ and zero otherwise. You need exactly one sample per $T$ seconds to represent $x(t)$, even though $x(t)$ is clearly not band-limited.

A smoothing kernel is just an optional filter that is applied to the continuous-time signal before sampling. For sampling a (quasi-)band-limited signal, the anti-aliasing filter would be an instance of such a smoothing filter.

An annihilating filter for a specific signal is an LTI system that produces an output of zero when that specific signal is applied at its input. E.g., for a sinusoidal signal with frequency $\omega_0$, a filter with a notch at $\omega_0$ is an annihilating filter.