That’s a interesting question and before I take a stab at it I’ll say I’m not a doctor but I have had allergies nearly 40 years now. So I have lots of experience, but not necessarially expert knowledge.
Rather than say “hyperactive immune system”, “mis-firing” might be more appropriate. There is some evidence (although the theory is not yet proven) that much of the IgE antibody system (the one primarially involved in allergies) evolved to deal with parasites, particularly intestinal parasites - which have been largely eliminated from the western world. Without nature’s intended target, in some individuals IgE apparently goes looking for trouble and latches on to stuff that is actually harmless. There is also some indication that parasites are capable of suppressing a host’s immune system, in which case parasties might be required, in a sense, to keep rampaging antibodies in check.
(Unfortunately, I’ve never seen any evidence that parasite infection AFTER allergies develop does anything to alleviate the allergies. Personally, my allergies are annoying enough that if adopting a tapeworm would help out I’d certainly consider the option)
Folks with mild allergies (which, fortunately, constitute the majority of folks with allergies) certainly suffer discomfort, but there’s isn’t much indication they have some sort of “super” immune system. Folks with severe allergies can be more suspectible than average to various illnesses, either becasue the body wastes resources against things that are basically harmless, leaving less for the real battles, or because the allergic reactions cause tissue damage that leaves an opening for invading bateria and viruses. People with eczema, for instance, may have multiple skin lesions and suffer from repeated skin infections. They are also more prone to certain potentially fatal reactions to vaccines and pharmaceuticals - google on “Stevens-Johnson Syndrome” for particularly gruesome pictures. The most fast and severe allergic reaction is anaphyllactic shock, which is frequently, if not usually, fatal if untreated and even with modern medicine can cause permanent disability in some cases. Being more immune to various germs is not much of a trade-off if accidental ingestion of, say, wheat or dairy products in minute amounts will kill you in minutes. Even in the case of mundane hayfever, the chronic congestion of untreated sniffles can provide a very nice environment for bacteria to start a sinus infection.
There is some - a little - evidence that, statistically speaking, allergic people are less likely to either develop cancer or to die from it. I’m not sure if they’ve taken into account deaths from allergic problems though, which may offset this advantage, if it even exists.
Allergics also tend to be more prone than average to auto-immune disorders, which would be in keeping with the whole concept of a mis-firing immune system in general.
Treatment with immune-suppressive drugs will definitely reign in those pesky allergic reactions, but the long-term side effects can be just as bad, if not far worse, than the allergic condition itself. Only in the most extreme cases are people put on long-term suppressive therapy.
At best, allergics may have better coping mechansims when parasitized (a feature largely unneeded in many parts of the world today) and their immune systems may be marginally better at ferreting out cells about to turn feral (cancerous). Mind you, a heavy enough parasite infection will sicken or kill anyone, and there are plenty of allergics on cancer wards, we’re talking about statistics involving large populations. Meanwhile, their allergies do make them suffer (not that evolution cares about the suffering of the individual, only the ability to reproduce) and may even, in extreme cases, kill them young.
As for fending off bacteria, viruses, and fungual infections - that part of the immune system doesn’t seem to have a very strong correlation with the IgE/allergy system. Someone could have extremely good health apart from their allergies, or they could be sickly all around. The immune system isn’t one entity, it’s a number of interlocking systems, any one of which may function better or worse than average.
This is further complicated by the fact that immune-suppressives ARE used to treat severe allergies and, of course, they suppress immunity. So someone with really bad allergies may be more sickly because of the medication required to control the allergies rather than because their entire immune system is defective.
In my own case, aside from the allergies I’m pretty healthy. I seem much more resistant than average to fungal infections (I’ve never had one - not even the yeast infections other women seem to have at least occassionally) and when I do get sick I recover quickly. However, untreated congestion has lead to some nasty sinus and ear infections in the past. Also, because I do suffer from eczema I get frequent rashes and usually have broken skin somewhere on my body at any given time. I have had repeated skin infections, and one of them required surgical drainage (the story of which is recounted in the infamous Pimple Thread). So while I recover quickly from infections, I also have more than average because a major defense system - my skin - is so often breached. Personally, I would have preferred to skip the rash and the infection in the first place, thank you very much.
I realize that was a rather complicated answer to your questions, however, it IS a complicated situation.