Hi,
Azolla does pull Nitrogen out of the air with help of symbiotic cyanobacteria Anabaena azollae, however if water already contains Nitrogen then it uses what's available instead of pulling out of the air. There has been conducted several studies on cleaning water of organic pollutants using Azolla and/or Lemna (mainy N and P) and it performs this task just as well or often better than Lemna (duckweed).
In ideal environment Azolla grows faster than Lemna in its ideal environment.
An ideal environment for Azolla differs from the one for Lemna. Azolla likes shade or partial shade and grows poorly in full sun, for Lemna its the reverse. Which one grows better in your case heavily depends on characteristic of the place you want to grow it in. In shade/part shade Azolla will thrive while Lemna will struggle or disappear, while in full sun it is the opposite. Azolla has 2 advantages over Lemna - it can pull Nitrogen out of the air if water does not contain enough and can shade Lemna as it is taller.
Perfect placement for Azolla ponds would be ground under photovoltaic arrays. PV's use the sunshine while shading Azolla pond underneath - double use of available land.
Even if Azolla is pulling Nitrogen out of the water (delivered in form of manure fertilization for example) and not out of the air, the Nitrogen stored in its tissues is more stable than the same Nitrogen directly out of the source (manure for example). If it is component of plant tissue, then when incorporated into the soil or dressed on top of the soil, it becomes food source for worms and microbes and holds onto the Nitrogen (and Carbon) in the soil. After the plant tissue decomposition, it is still bound to organic mater in the soil which does not volatize so readily as mineral Nitrogen. It is then made available to plants by microbes "per order" via mechanism of root exudates.
Barrels are not the best containers for growing neither Azolla or Lemna. Both require surface area and do not care about depth (around 20-30cm is plenty). Better way to utilize barrel is to cut in in half lengthwise - that way you multiply the surface area several times compared to barrel stood up vertically.
Or use containers like old bathtubs or fridges.
I prefer to make shallow ponds (around 20cm depth) by either digging a bit of soil and making mound around the pond or making the rim out of logs or concrete blocks. In both methods such shallow pond is lined with cheap polypropylene sheet and filled with water. Quick to make in nearly any shape or size you want and costs next to nothing.
Around 20cm is enough. If you are often going away or you are forgetful and where you live there is not much rain in the summer then its better to make it a bit deeper. That way replacing evaporated water can be more sporadic and you have more time to do that before the pond dries up completely killing the culture.