For example, if we have two frames that have constant relative velocity, would the interval be the same between any two events? Or should we impose the condition that the particle must not accelerate between these two events?
The spacetime interval between two events is always the same in any two inertial reference frames- that is, as long as the frames themselves are not accelerated. It doesn't matter if these events are points on the worldline of an accelerating particle, a particle of constant velocity, or just two events picked from spacetime at random.
Events do not "occur" in frames. They are measured or described relative to frames, but the events themselves occur in spacetime. For example the emission of a particle and absorption of that same particle later are events. Different inertial frames would assign different coordinates to these events, but the spacetime interval between them is the same in all cases. What the particle itself does between events is irrelevant to the coordinates assigned.
Non-inertial frames involve coordinates that are accelerating, that is, the coordinates themselves are changing, and so for them the interval is not, in general, invariant. For example, the metric in Rindler coordinates with constant proper acceleration $\alpha$ in the $x$ direction is $ds^2 = -(\alpha x^2)dt^2 + dx^2 + dy^2 + dz^2$. Such coordinate systems are harder to work with than inertial frames.