The DEET/carcinogen thing is urban myth and pseudoscience. (As best we know at the moment, that is...) I see that meme pop up with some frequency, and try to squash it where I can. It has an incredibly long and safe track record. CERTAINLY the risk of malaria, West Nile, Lyme disease, RM Spotted Fever, etc. FAR outweighs the risk from DEET.
DEET is primarily meant to be applied to skin, though assuming that your clothes didn't melt you would of course still benefit from applying it to clothing. Permethrin OTOH is meant to be applied to clothes, and lasts through several washings unlike DEET. I am incredibly familiar with both, being a military physician (gotta love those preventive medicine classes) and also having looked up a bunch of scientific articles about DEET a year or so ago during a similar forum discussion.
Yes, it is toxic (NOT carcinogenic) in large amounts. LARGE amounts. As with many insecticides and repellents it has a nerve-gas-like ability to inhibit acetycholinesterase. You'll know if you're using too much because you'll get twitchy kinda like you overconsumed caffeine (though the mechanism is somewhat different from that of caffeine). By far most cases of significant toxicity involved idiocy on the part of the user- and, yes, usually involved 100% DEET because it is easy to abuse in that concentration. I think a child died after drinking some, and it is possible that it might contribute to seizures in susceptible individuals. I recall that there have been a handful of fatal seizures in which DEET may have been a contributing factor- which is not an unreasonable hypothesis given its neurologic activity- but no one is really certain. So if you have a seizure disorder you should avoid it.
I, personally, stick to 'military-grade' 33% DEET.
Also, yes it will dissolve some plastics and other synthetics but generally once 'dried' on the skin you needn't worry about it rubbing off and melting something. Rayon, nail polish, and varnishes in particular are susceptible. I still wouldn't touch plastic eyeglass lenses, though, as I imagine that might be enough to form a hazy fingerprint etched into the lens. I have seen a pair of old Army RPG glasses melt when DEET was applied directly to them.
Don't let naturalpathic people scare you away from DEET. Yes, it is not very 'natural', so if that's your thing by all means avoid it. But it is also the only skin-safe insect repellent that has been proven to work well, notwithstanding the citronella and eucalyptus fanboys. (If nothing else, the natural stuff needs to be re-applied essetially continuously to have any effect.) If the natural stuff works for you, all the more power to you, Brother. But you simply can't beat DEET for effectiveness if you aren't otherwise opposed to it.
I have heard of another synthetic repellent called Icaridin that may have promise, but I'm unsure about studies verifying its effectiveness.