How complete is the information in DNA?
This is a good question, because the answer to the overall question (is the information in DNA complete - yes) is not the same as the one to the more specific ones (would a cell survive if the DNA was swapped with that of another life form - no). Let me elaborate:

The information in the DNA of in any given organisms (except for a virus) IS complete. This means that the DNA has 'code' for all the molecules required to build and sustain this organism.

However, the DNA of an organism alone (or placed inside an unnatural environment like a completely different cell) is not sufficient to actually make or build that organism. The reason for this is that, while the DNA has all the necessary code, without the correct machinery to actually read that code nothing can or will happen. Eukaryotes and bacteria are so vastly different (which is to be expected after millions of years of evolution for both), basically all the different machineries are not compatible anymore: transcription binding sites are different, gene structure (exons/introns vs. operons) is different, Ribosomes are different, ... [btw this also means that just putting the gene of an eukaryote into a bacterium doesn't work, researchers have to modify it accordingly to work].
Even between different eukaryotes (and probably some bacteria but I'm not sure about that) exchanging the nuclear DNA would not produce viable cells (or at least not ones that could grow into an organism), since again the transcription machinery would not match up sufficiently.