We know from quantum mechanics that microscopic particles have spin, which is a kind of intrinsic angular momentum. The particle has angular momentum without physically rotating. In a similar way, do particles also have intrinsic linear momentum? That is, can the particle have linear momentum without physically moving? If not, then why not?
Do particles also have intrinsic linear momentum (linear analogue of spin)?
In addition neutrinos travel at a speed which is so close to c that we can't tell the difference. And the neutrino "was first hypothesized by Wolfgang Pauli in 1930, to account for missing momentum and missing energy in beta decay".
We know from quantum mechanics that microscopic particles have spin, which is a kind of intrinsic angular momentum. The particle has angular momentum without physically rotating.
You need to be cautious here, because the Einstein-de Haas effect demonstrates that "spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". And because the Wikipedia electron magnetic moment article says this: "From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet". A bar magnet has a magnetic field like a solenoid, and you can simplify a solenoid to a current loop. In addition check out the static-fields section of the Wikipedia Poynting Vector article:
Note where it says this: "While the circulating energy flow may seem nonsensical or paradoxical, it is necessary to maintain conservation of momentum". IMHO you need to think in terms of a tornado. It has an intrinsic spin, and this makes it what it is. The air is rotating, but the tornado itself isn't. If you could somehow stop the rotation, it wouldn't be a tornado any more, it would be just be wind.
In a similar way, do particles also have intrinsic linear momentum? That is, can the particle have linear momentum without physically moving?
No. A particle such as an electron can have an intrinsic angular momentum without physically moving.
If not, then why not?
Because when it has linear momentum it's moving linearly. However what Slereah said is kind of right: "the equivalent of intrinsic spin for linear momentum is the mass". You can understand this by thinking of the photon in the mirror-box. See http://arxiv.org/abs/1508.06478 (the 't Hooft is not the Nobel 't Hooft). When you catch a massless photon in the box, it increases the mass of the system. The photon is still going round and round at c, but it's effectively at rest, so the momentum is now exhibited as mass. Photon momentum is resistance to change-in-motion for a wave propagating linearly at c. When the wave is going round and round at c, we call its resistance to change-in-motion mass. This is what underlies E=mc². When you open the box, it's a radiating body which loses mass. In a way mass is just the flip side of energy-momentum.