Does anaerobic respiration involve the breakdown of nitrate?

No, both aerobic and anaerobic respiration involves the breakdown of (usually) glucose. However, anaerobic glucose is simply converted directly into lactic acid, while aerobic respiration gives time for glucose to be completely broken down into acetyl-coA and filtered through the Krebs cycle. Therefore, to get the same amount of energy, anaeorbic respiration has to use a lot more molecles of glucose. Other sources of energy are fats and proteins.