I'm not sure what the rod might have been, the inside of an air tank is pretty simple.
Replacement drain valves are available in a variety of configurations, and generally they thread into a nut welded into the tank. It's then just a matter of finding an appropriate replacement valve (relatively easy) and removing the old one (probably much harder).
![replacement valve](https://i.stack.imgur.com/fzzlH.jpg)
To help get the frozen valve out, you could (should) try penetrating oil to free what are almost certainly frozen threads, but you may need to step up to using a torch to apply heat to help break the part loose.
You are correct that it is a good idea to use teflon tape on the threaded fittings, however no sealant is required or advised on the quick-connect part itself. These are designed to seal using internal o-rings.
There are multiple types of quick-connect air line fittings that at first glance appear similar, but are not necessarily compatible. It is very likely that you have connected two incompatible types, resulting in a leak. This is very common.
The two "main" types are often referred to as "automotive" and "industrial", though each can have sub-types that are typically letter designations, e.g. Industrial can be M, H or G. There's also a type called ARO, and one called Lincoln, but these are somewhat less common (at least in the US consumer market).
![air fittings](https://i.stack.imgur.com/QDCNP.png)
(image from Legacy Manufacturing)
The documentation for your air compressor may specify exactly which type of fitting you have. Otherwise head to the local big box store with your male coupling that leaks, and look for a similar-while-slightly-different male coupler to try.
Legacy manufacturing has published a very informative guide on the evolution of the various types of quick-connect fittings that lists the common types in use, with some images of each.
Milton Industries has also published a document that may help identify the type of plug you have, though their letter designations may not exactly match those of other manufacturers.
Best Answer
Check to make sure one of the (presumably plastic) panels that make up the exterior of the unit are not loose. My unit had a "fancy" black plastic border that fastened around the front panel, and it would over time begin to work loose as the compressor vibrated. I ended up wedging a little folded piece of paper between the front panel and the bordering plastic to get it to shut up.
Other than that, put something vibration-absorbing under it if possible, as Ed Beal suggests. There's not a lot to be done. These units are noisy because the usual outdoor bits are now in a box next to you. But if you need cool air, you need cool air!