Human genome, chromosomes
Do all chromosomes have the same DNA?


No.  Not any more than each chapter of a book has the same words in the same order.


  If so, does it mean that in different cells with different functions
  some genes are expressed and some aren't?


Yes.  In general, all cells in your body have all the same DNA; what makes your retina cells function differently from your pancreatic beta cells is which genes they express.  Like if every chef in a restaurant had the same recipe book which had every recipe the restaurant can make, but the pastry chef only makes pastries.


  Also, if each chromosome has different lets say "fragment" of the
  genome, how is the genome interconnected?


The genome is not physically connected except for all the genes on a chromosome being physically connected.  But a gene product from one  chromosome might control expression of another gene by physically binding to the DNA close to it, which can affect its expression.  But most gene-gene interactions are really protein-protein interactions.  (Or in some cases, RNA interacting with RNA)