The main references are Deligne’s three papers
The first paper is simply a summary of Deligne’s ideas on the relationship between weights in Hodge theory and weights coming from the action of Frobenius on etale cohomological in positive characteristic. Note that in the last section, Deligne simultaneously formulates the weight-monodromy conjecture (still open) and his expectations on the existence of limit mixed Hodge structures (established by Schmid — see below).
The second paper uses various results on logarithmic poles and residues which may be found in II, §3 of
Deligne, Equations Différentielles à Points Singuliers Réguliers
In the third paper, Deligne crucially uses the method of cohomological descent, with references to SGA4. For a very readable account (which essentially avoids SGA4), see
Conrad, Cohomological descent
The existence of limit mixed Hodge structures was conjectured in Deligne’s first Theorie de Hodge paper, as mentioned above. It was established (in greater generality than conjectured by Deligne) using analytic methods in
Schmid, Variation of Hodge Structure: The Singularities of the Period Mapping
An algebraic account may be found in
Steenbrink, Limits of Hodge Structures
or in
Peters-Steenbrink, Mixed Hodge structures
The limit mixed Hodge structure can provide interesting invariants of singularities. The case of isolated hypersurface singularities is explored in
Steenbrink, Semicontinuity of the Singularity Spectrum
A nice write-up on the spectrum of hypersurface singularities is
Van Straten, The Spectrum of Hypersurface Singularities
The paper which describes the Thom-Sebastiani formula for the spectrum is
Scherk, Steenbrink, On the Mixed Hodge Structure on the Cohomology of the Milnor Fiber
For a variation on the results and methods of Theorie de Hodge III, see
Du Bois, Complexe de de Rham filtré d'une variété singulière
Mixed Hodge structures may have nontrivial extensions; in fact, groups of extensions are typically quotients of complex vector spaces by discrete subgroups ("generalized complex tori"), see Proposition 1 of
Carlson, Extensions of Mixed Hodge Structures
which also proves a Torelli-type result for singular curves. On the other hand, all higher Ext groups vanish in the category of mixed Hodge structures — see Corollary 1.10 of
Beilinson, Notes on Absolute Hodge Cohomology
which contains many other interesting results.