What are the key differences between a particle and a wave? When do we call something a particle, and when do we call something a wave? Do we call something a particle when it is highly localized? If that's right, then a wave packet is a particle. Do we call something a wave when it is not localized and when it shows interference pattern?
 A: If a state is localized in position basis then it is called a particle (something that has a definite position) and if it is localized in the momentum basis then it is called a wave (something that has a definite wavelength, which, in Quantum Mechanics, relates to the momentum).
Having said that, I would add that in practice, any quantum state that is fairly distributed in the position basis is called a wave. It is not a strict definition. Moreover, it should also be noted that what we call the wave-function (i.e., the position basis representation of the quantum state) is not a wave in a mathematical sense, i.e., the equation according to which it evolves in time (the Schrödinger equation) is not a wave equation, it is a diffusion equation. So, to be honest, we simply mean a distributed or a little bit wiggly pattern when we say "wave" in the context of Quantum Mechanics. A more precise definition would be a distribution that has a specific wavelength (i.e., a specific momentum) as I mentioned, however, even such a wave-function wouldn't follow the wave-equation as it evolves, it would follow the Schrödinger equation (i.e., a diffusion equation). 
Finally, all quantum states exhibit interference. However, you are correct in noticing that particle-like states wouldn't exhibit any distinctive interference-like behavior in the position basis. But, any wavefunction that has a distributed or a little bit wiggly pattern would exhibit noticeable interference, in particular, one doesn't need a distribution with a specific wavelength to exhibit interference. 
A: 
Do we call something a particle when it is highly localized?

yes

If that's right, then a wave packet is a particle.

In quantum optics, photons are often represented as localized "wave packets". The "field theory" method of determining if something is a "particle" is if it obeys a particular commutation relation. Often times this leads concluding that in some interacting systems, so part of the system has this particular commutation relation and it's called a "quasi-particle". For example, the interaction between atomic dipoles and photons are labeled "polaritons" and are considered a type of "quasi-particles" because of their commutation behavior:
$$\left[\hat{\Psi}_k,\hat{\Psi}^+_{k'}\right]
 =\delta_{k,k'}\left[\cos^2\theta
 +\sin^2\theta\frac{1}{N} \sum_j(\hat{\sigma}^j_{bb}+\hat{\sigma}^j_{cc}) \right].$$

Do we call something a wave when it is not localized and when it shows
  interference pattern?

Yes. Although this is the trickiest to answer. Wave are things that have multiple values distributed across space and follow some sort of differential-equation that looks like a wave. I'm careful not to say that "waves follow the wave equation" because quantum waves do not follow this (they follow something that behaves pretty "wave-like" resembling a complex version of the heat equation). 
Electrical waves follow the wave equation (through a simplification of Maxwell's equations).
Classical particles (in some materials, like water or air, for example) are held-together with a particular force (a harmonic oscillator usually) and can be shown to follow each other in a way that follows the wave equation.
Quantum waves follow Schrödinger's equation.
Photons follow both the wave equation and Schrödinger's equation.  
