Why do soil bacteria produce nitrous oxide as a result of anhydrous ammonia fertilizer application?
Addressing the first part of your question:


  Is it common for most soil bacteria to make significant quantities of
  nitrous oxide after ammonia is added, or is there only a small subset
  of bacteria present in soil that might do this?


No, not all soil bacteria can oxidize ammonia. According to wiki:


  The oxidation of ammonia into nitrite is performed by two groups of
  organisms, ammonia-oxidizing bacteria (AOB) and ammonia-oxidizing
  archaea (AOA).


The role of archaea in the soil nitrogen cycle is not well-known yet, although it could be greater than that of bacteria. On the other hand, the role and metabolism of AOB are more or less studied.

The process of ammonia-oxidizing is also called nitrification and AOB sometimes referred as nitrifying bacteria. The most known bacteria genera from this group are Methylomonas, Nitrosococcus, Nitrosomonas and Nitrosospira. I think the most studied species is Nitrosomonas europaea Winogradsky 1892.

For reference see: Arp DJ, Sayavedra-Soto LA, Hommes NG. Molecular biology and biochemistry of ammonia oxidation by Nitrosomonas europaea. Arch. Microbiol. 178 (2002) 250-5. 

The second part of the question:


  What is the metabolic pathway in this case?


Although, the metabolic pathways vary in different genera, let's look at ammonia metabolism in N. europea.

In the simplest case, it is a two-step process.
According to KEGG:

Step 1. The enzyme ammonia monooxygenase converts ammonia to hydroxylamine. The enzyme contains copper, iron and possibly zinc. It requires two electrons, which are derived indirectly from the quinone pool via a membrane-bound donor. The reaction is:

NH3 + a reduced acceptor + O2 = NH2OH + an acceptor + H2O


Step 2. The enzyme hydroxylamine dehydrogenase converts hydroxylamine to nitrite. This reaction requires specialized cytochrome as an acceptor of electrons. The reaction is:

NH2OH + H2O + 4 ferricytochrome c = NO2 + 4 ferrocytochrome c + 5 H+


While nitrite is the main product, the enzyme from N. europaea can produce nitric oxide as well. Therefore Step 2 can produce nitric oxide (NO). In this case, additional mechanisms of converting NO to NO2 are involved.

Further destiny of released nitrite can be different. In one process it could be oxidized by different microorganisms (bacteria, archaea) to nitrate (NO3) which could be consumed by plants or washed out to deeper horizons with water. In another case, nitrite, nitric oxide and nitrate could be involved in process of denitrification. During this process, a group of denitrifying organisms (bacteria, archaea and fungi) consume nitrogenous compounds and reduce them to nitrogen (N2), which can escape to the atmosphere. This process can occur through different metabolic pathways mainly with aid of reductases like nitric oxide reductase. One of the possible intermediate product of such pathways is a nitrous oxide (NO2). If N2O escapes from cells we can observe a resale of nitrous oxide from soil.

Let's return to your first question, and particularly to the part where you ask about significant amounts of nitrous oxide.

First of all, nitrification is a natural process and is a part of the nitrogen cycle. Some organisms fixate nitrogen and convert it to ammonia, some oxidize ammonia to nitrous compounds and some of them reduce it back to nitrogen. It occurs everywhere virtually in all terrestrial and water biocenoses. In a healthy ecosystem, the inflow of nitrogen is theoretically equal to the outflow (with temporal deposition in trophic chains). In such natural conditions, the release of nitrous oxide should be neglectable.

The situation changes significantly when we add an ammonia to soil artificially as a fertilizer. The main problem is that we break an equilibrium between nitrogen fixation, nitrification and denitrification. The second big problem is that an ammonia is toxic for most organisms, so we can alter microbiome. In such case, we can expect an excess of products of nitrification and improper denitrification with exaggerated levels of released nitrous oxide. The actual proportions vary significantly depending on both biotic and abiotic conditions, therefore it is difficult to predict real levels of nitrous oxide released without in situ examination.

P.S. sorry for possible bad writing.