Yes, at least as far as co-association in the sediments allows us to make this conclusion. In the Holocene deposits of Bear Lake, Utah XFRD results indicate that aragonite and calcite have alternated over time with changes in lake size. During small lake phases, the total dissolved solids concentrations increase to the point that aragonite and gypsum precipitate. Bear Lake is deep enough that the bottom waters are often anoxic.

In a shallow lake, you could easily find the same conditions if the lake becomes meromictic. If there is enough sulfur available, pyrite and pyrrhotite may form in the monimolimnion, while gypsum forms in the epilimnion. 

Dear Dr. Mishra,

the presence of gypsum and pyrite may simply be predicted by the physical-chemical regime irrespective of the environments of deposition, be it lacustrine or any other freshwater environment.

1. The precipitation of gypsum and pyrite  is controlled by the availability of bivalent Fe and calcium in combination with sulfide or sulfate anions. They precipitate as the solubility product is exceeded.  Pyrite is pecipitated when the Eh is < 0 and gypsum needs an Eh > 0.

2. Yes, if you add, e.g. , Ca(OH)2 to a sulfate-bearing system you will get CaSO4 . 2 H2O;  it does not depend upon the state of evaporation.

3. In many waste dumps at mining sites you can encounter  pyrite being converted into gypsum provided calcium ions are present. The climate needs not be aridic .

You can model it simply by using x-y plots ( x = pH, Y = Eh given in mvolt) with the soluble components being sulfur, iron and calcium in their ionic state.

Best regards

H.G.Dill

An on-going study of Late- Creaceous black shale in Nigeria shows gypsum occuring. If gypsum precipitates in oxic conditions then I shouldn't expect to find it in my organic rich shale.