Can anti-Higgs particles have formed before any baryonic matter existed or is the Higgs boson always its own anti-particle? After reading the book Das Dunkle Universum by Adalbert Pauldrach ( which btw is very well written) questions remain regarding the Higgs field and it's role in the early universe. According to the hypothesis by Tulin and Servant, initially Higgs and anti-Higgs Bosons formed. Since the former interacted and produced mass particles, anti-Higgs Bosons had no more partners to interact and are sitting around as dark matter. 
Question: Can anti-Higgs particles have formed before any baryonic matter existed or is the Higgs boson always its own anti-particle ? Is it plausible that a particle with a half-life of $1.56 × 10^{−22}s$ can stay around for 14 billion years because it has no reaction partners ? 
Regarding the Higgs field: does it have the size of the universe or are there innumerable individual fields in the universe?
 A: The relevant paper is Higgsogenesis by Geraldine Servant, Sean Tulin, Phys. Rev. Lett. 111, 151601 (2013).
I suspect you have confused the Higgs boson with the Higgs field. The Higgs boson detected at the LHC is an excitation of the Higgs field that exists only at energies below the electroweak phase transition. The paper by Servant and Tulin discusses interactions occurring at and above the EWPT energies where the Higgs boson does not exist.
As for the paper, from a brief scan through it looks plausible but is just one one of many ideas that have been suggested in this area. The paper hasn't been extensively cited so it appears not to have generated very much interest.
A: I will address this part of your question:

Regarding the Higgs field: does it have the size of the universe or are there innumerable individual fields in the universe?

The Higgs field and the bosons associated with it are within  a quantum field theory model.  The standard model of particle physics is the starting point, and then there are extensions and modifications, all within quantum field theory. 
The standard model of particle physics is based on the basic elementary particles listed in the table, and the postulate is  that all matter and energy are dependent on them. In  quantum field theory, each of the particles in the table 

covers the whole space time with the wavefunction of the free point particle field, which is the quantum mechanical solution of the appropriate equation ( the Dirac for the electron as an example) . Acting on these fields are creation and annihilation operators, which  generate the particles as excitations of the field . In particular, calculations are possible to derive interaction crossections and decays using perturbative expansions and the appropriate Feynman diagrams.
As you see there is a limited number of fields, and the Higgs field , one only in the standard model  , is also covering the whole spacetime. In the standard model the Higgs is its own antiparticle, there are various extensions and modifications which have more than one Higgs field and more elementary particles than the standard model table , possibly invoked when trying to explain open questions in cosmology, as  baryon asymmetry and dark matter.
John has covered the rest of your question.
