Is the particle entanglement a boolean property? That is, when we consider two given particles, is the answer to the question "are they entangled" always either "yes" or "no" (or, of course, "we are not sure if it's yes or no")? Is there such thing as partial entanglement?
No, it's not a Boolean property. Entanglement between two quantum systems (could be particles, or anything else) could be partial, and can be quantified using different measures. In the specific example of Bell states, the two systems (each of them with 2 states $|0\rangle$ and $|1\rangle$) are said to be maximally entangled with the entanglement entropy being 1 qubit.
Yes, the answer is always either yes or no. Being entangled means that they are not in a separable state. A state is called separable if and only if it can be written in the form $\rho = \sum_i p_i \rho_i^A \otimes \rho_i^B$, where $\rho_i^A$ refer to the state of particle $A$ and $\rho_i^B$ to $B$. This is always either the case or not the case, regardless of how much entanglement precisely we assign to the two particles. All standard entanglement measures satisfy the property that $\rho$ is entangled if and only if $E(\rho) > 0$ (ie. $E(\rho)=0$ for all separable states).
To find out whether given two particles are entangled, people have developed procedures known as entanglement witnesses, and research into the entanglement witnesses has formed a sub-discipline in quantum information theory. Bell inequalities can also be thought of as entanglement witnesses since their violation implies the presence of entanglement.