Likelihood for a model

Question

I am not able to understand what is $x_0$?

Answer 1

The model apparently has some observables. In this case, you've made an observation and gotten the specific value $x_0$. Now it's asking about the conditional distribution $f(\theta|x_0)$ given the information in the observation.

Answer 2

Regarding the second question - the answer is no. Take for instance $U[0,\theta]$ where $\theta \in [0,1]$, then $$ \int_{[0,1]}L(\theta|x_0) d\theta =\int_{[0,1]}\frac{1}{\theta^n} d\theta, $$ is not even integrable over $[0,1]$ (Actually, $L$ is basically not defined for $\theta=0$, so you can take $(0,1]$ for better ollustration. In this case, the integral is simply diverge).

The likelihood function is a function defined over the parametric space $\Theta$, i.e., it is not a density w.r.t to $\theta$, hence, integration w.r.t the parameters can equal whatever you want or can be even non-integrable at all.