The dynamics changes and the exact identification of the key soil water and N processes under any treatment may be numerous so It would help if you could narrow the scope of your questions a bit. I assume, you have limited resource. If that is the case, you may go after the usual rate limiting processes e.g., decomposition, mineralization and possibly nitrification. By studying these processes you'd also gain information regarding available N (ammonium, nitrate and possibly amino acids). Target techniques for these approaches vary from simple in situ extraction and inexpensive elemental analyses to more complex methods involving stable isotopes. For water, either gravimetric or volumetric measurements of soil water content should give you meaningful information unless you are comparing the effects of organic fertilizers in soils with different texture. Finally, you could consider examining these dynamics indirectly by looking at the effects on bio-indicator particularly plant i.e., how they grow, their tissue accumulation of N, and their water relations. Once you have narrowed your questions, I am confident many more researchgaters will step in to respond to your specific questions.
One point which has not been covered is various path ways of N losses especially through denitrification (N2O and N2 gases),volatilization( losses of ammoniacal N ) and leaching losses of nitrate N.From climate change point of view the gaseous forms of N emissions especially through denitrification are important.
Dr Waqas , while studying the pathways through which nitrogen goes out of the system , like denitrification , volatilization and leaching , the these will remain the premier ways of N-losses. however , while practicing organic fertilization , one process could dominate over another process , but the principal ways will remain the same , with an eye on process of immobilization .
As rightly explained by Dr. Bassirirad, it is subject on which so many books and thousands of papers have been written and still a lot of uncertainity remains. You should narrow down the scope of your question. Take into consideration the nature of soil, crop, water management and N management and then proceed step by step to reach at some conclusions after going through pertinent literature. Your question cannot be answered in a paragraph or two.