If that coil is in the 80 and 40 meter dipole, then it isn't a simple coil, it's a trap. If it's a trap then it has not only a coil, but a capacitor, making it a tuned circuit. That tuned circuit is tuned for somewhere in the 80 meter band (bottom end) and presents a very high impedance to a 40 meter signal, thereby acting as on open switch cutting the wire off at that point. Just like any other 'trapped' antenna.
There are a couple of things that lead me to believe that that coil is just that, a coil, or loading coil. From what Mole' says, there's not much wire on the outer end of it, so it's not 'enough' length for 80 meters, so that coil is providing the needed inductance to make up for a shortened antenna. Another thing is that there is no apparent capacitor associated with it, so it isn't gonna act like a trap.
I think you might wanna re-think the thing about which/what the other 'elements' in that antenna are for. Keep in mind that 'even' multiples of harmonically related lengths are 'bad', but 'odd' multiples are 'good'. (For example, an 80 meter dipole is terrible on 40 meters, but works out well on 15 meters. "Works well" is relative, certainly doesn't mean is better than, or equal to a 15 meter antenna, just 'workable', sort of. Terrible way of putting it, ain't it?)
The 'Alpha-Delta' series of antennas work with harmonically related segment lengths. Some are shortened with loading coils. I honestly can't remember ever seeing one that used a trap (which doesn't mean a lot since I certainly haven't seen them all).
(By the way, I'm doing as fine as possible considering the conditions. Which means that there are somethings much worse than they used to be, and something I can't tell any differences in. Not sure if my memory is in that first group or the second... Oh well, I'm having fun so who wants to worry about it, right? Went back to work, found I like it. Not the 'work' part, but the part about 'getting out of the @#$ house'! They even pay me!)
Oh, one last thing.
Baluns are great for their intended purpose, making the change between a balanced antenna and an un-balanced feed line, which can make a difference. They also have some very limiting characteristics, as in they are designed for use at particular frequencies. Get much away from those design frequencies and things go to 'pot' very quickly. I've found that for typical HF
use, they just are not all that necessary, and are definitely NOT a 'cure-all'. Biggest difference is that using them or not using them affects the radiation pattern more than anything else. For antennas that depend on a particular radiation pattern, changing it ain't exactly the best idea in the world. For a typical dipole, they can certainly be done away with and not really be missed. As far as impedance transformation is concerned, they are misused to a very great extent. If you just 'gotta', then use a 1:1. Depending on a huge number of things, most people have no idea of what the actual impedances involved are (what they are supposed to be, sure, but the key words there are "supposed to be"). Oh well. If it works, good. If not, try something else...