What is 'Retention period' in chromatography and what is its importance?
To answer your three questions in series:

1) The time is a measure of how strong the interaction of a compound is with the column used. The stronger the interaction the longer the compound will stay on the column and this is therefore a way to separate two compounds. For example one molecule with a weak interaction spending a minute on the column, the other spending 5. Collecting fractions of solvent from a liquid chromatography column will yield pure compounds. Note that the time is also a measure for the amount of 'solute' flushed over the column. Many chromatograms also report retention in column volumes of solute. I will not go into detail about that here. It might be worth a separate question.

2) If no peak is found, no retention time can be reported. In chromatography it never (actually almost always) good when a compound fragments/degrades. That means the technique is not suited for purification or analysis. In unique cases, the degradation can be used to analyse whether or not a compound will degrade when interacting with certain materials.

3) In mass spectrometry a retention time in the machine is, as far as I am aware a 'Time of Flight'. Compounds are quickly brought into a gaseous and charged state and are pulled through a magnetic field. Depending on the weight of a compound, the time it flies, is shorter or longer. That is a way of calculating the weight of a molecule. In some cases, for example small proteins, the molecules are purposefully fragmented. The fragments can then be used to reconstruct of what amino acids the protein is made of.

Hope this answers your question! If any clarification is needed, please, do ask :)