According to Cambridge University, the general terminology is that a "teleconverter" is mounted on the rear of the lens and a "teleconversion" filter is mounted to the front of the lens. In both types, there will be some degrading of the image quality and the degree of the degradation usually varies with the price. The cheap ones from China will have a very definite negative effect on image quality. On the other hand, those made by the camera manufacturers can be quite good with only a small negative effective on image quality. Both types work best on prime lens, since they will magnify the softness typically found in the image quality of zoom lens at the long end, but the sharper the lens, the better the results whether prime or zoom.

An advantage of the filter type is no loss of light, so a 2X teleconverter type will lose 2 stops of light (a 100mm f2.8 becomes a 200mm f5.6), while a 100mm f2.8 lens with a 2X teleconversion filter becomes a 200mm, but will still be f2.8.

A disadvantage is the teleconversion filter is rather heavy on the end of the lens and can stress the autofocus mechanism. They are also quite large in diameter as someone else has mentioned.

I have a Canon 1.4X and a Sony 1.7X that will fit on any lens with a 62mm filter size or can be used with a step down adapter ring. Both are very high quality and I use them on a Minolta 135mm f2.8 for low light use and on a Tokina 400mm f5.6 to get extra reach when needed. There is very little loss of image quality with either. Since they can be relatively quickly screwed on to the end of a lens, they do not potentially introduce dust into the camera as a rear mounted teleconverter might during mounting and they can be carried in a coat pocket. I have tried several other teleconversion filters from China and/or Hongkong and results were very poor. I found both the Canon and Sony on Ebay and nobody was bidding on them, so I bid and got both quite cheap. I think the Canon 1.4X went for $0.99 plus shipping and is probably the best buy I have ever made on Ebay. The ones from China, however, would be a bad buy even at $0.99 in my opinion. The camera manufacturer types are rare on Ebay, but based on the two I have, I would definitely consider them. I think the Canon originally sold for over $200 and is no longer made. Sony still makes one that is smaller than the one I have and it sells new for $100 to $200 as I recall, but it is too small to fit on the end of a 300mm-400mm lens with out vignetting, due to the required step up adapter. The larger 62mm one is no longer made.

This gets quoted so often on here, and it is very misleading. A jpeg is not an image, any more than a raw file is... Try looking at it without a computer! Both jpeg and raw files are just collections of 1s and 0s, until some software program translates them into a viewable image.

I understand and agree with the concept that jpeg files have been pre-cooked, as it were, and will appear very different to an unprocessed raw file. It's just this idea that jpegs are already an image while raw files are not which needs to be straightened out. They are both just different types of data storage...

Extenders work best with fast prime telephoto lenses. They are pretty good with fast tele zooms. The 80-400 is neither. At 400mm the quality is barely acceptable. Through an extender on it, sure it will work - but if your intent is to get closer with mm to bet more detail, you're not going to get it. And you will have to contend with more noise from needing higher ISO and slower focusing. Fast primes and pro-level zooms are extremely sharp, so a 5% loss of acuity is manageable but the 80-400 compared to a 400mm prime or even the 200-400 prime isn't even in the same ballpark. Putting a TC on it won't get you what you are looking for.

BTW, what are you using for support of your planned 640mm lens (or longer if you are on a cropped sensor)?

A ball head is pretty much a waste of money on a monopod with ANY lens, and absolutely a waste with a long lens that has a tripod collar. You need a simple tilt head on a monopod as "portrait" mode will be attained via the tripod collar rotation, and "panning" is done with the monopod itself.
Any monopod with a load rating of 11 pounds or more will safely handle your setup. A tilt head from Manfrotto, RRS, Kirk or Shoot will work extremely well with the tripod and they are all lightweight, but the prices will vary by a lot between them.