In https://arxiv.org/abs/1407.2605, it is argued that all photons are necessarily circularly polarized, and linearly-polarized photons must be a superposition of such circularly-polarized photons. Besides, it and suggests that Bell experiments may really be measuring correlations of photon pairs.
My question is, do all Bell tests assume linear polarization, and will the Bell's theorem be invalidated if all photons are really as described in the mentioned paper?
Circularly polarized photons are linear combinations of linearly polarized photons, and vice versa.
The "classic" exposition of a Bell test assumes the testing of linearly polarization, but one can imagine other tests as well.
The paper you refer to is from a pair of interesting chaps, but I don't think it is a peer reviewed paper.
This answer elaborates on the answer by Paul Young.
For photons, any linear polarization can be expressed as a superposition of circular polarizations, and any circular polarization can be expressed as a superposition of linear polarizations. This does not affect the significance of Bell inequalities or the fact that they can be violated in the real world. Any single-photon polarization-state can be expressed as a superposition of any two distinct polarization-states, and any two-photon polarziation state can be expressed as a superposition of any four distinct combinations of two single-photon polarizations.
Bell inequalities refer to correlations between measurement outcomes when various observables are measured. The important things to consider in this context are (1) what observables are being measured? and (2) how was the state prepared prior to the measurement?
Many different observables relate to the polarization of a photon. One observable corresponds to L/R-circular polarization, one corresponds to V/H-linear polarization, and there are infinitely many others. We can measure any one of these at any given time. No matter what polarization the photon had prior to the measurement, if we measure the V/H-linear polarization observable (say), then the answer will come out either V-linear or H-linear. This is essentially the definition of "measuring" an observable: a measurement is any interaction that effectively projects the state onto one of that observable's eigenspaces. (Please note the word "effectively" here; this is a pragmatic statement, not a statement about how quantum theory should be "interpreted.")
In a prototypical example of "entanglement," a pair of photons is prepared in such a way that a Bell inequality is violated when particular combinations of polarization-observables are measured. The so-prepared state can be described either in terms of L/R circular polarizations or in terms of V/H linear polarizations; the two descriptions are equivalent, and how we describe the state has no bearing on the measurement outcomes.
