Language / Duzy / wielki - synonyms? [44]

duzy is like 'big, large'
wielki is 'bigger than big' :-) so it's more like 'great, huge' and is slightly less formal. You use it rather to express your own opinion about things (you can also say 'olbrzymi, potężny') and to emphasize not only size but also the importance, impression or feelings.

Language / The most common Polish case used in everyday conversations? [10]

I'm afraid all cases can be mixed within one sentence, for example we can say: Marek (nominative) dał Gosi (dative) książkę (accusative) do angielskiego (genitive). (Marek gave Gosia an English handbook) - If it's what you meant in your question about cases.