What are the benefits of no chill brewing? Are there downsides to the method? I am Specifically thinking of the no chill cubes.
The downsides as far as I can see are (in descending order of severity)
1) Possible toxicity of the plastic.
2) Recipe alterations due to the continued elevated temps after the boil is over.
3) Risk for contamination.
Right now, the current thinking is that "food safe", "heat resistant" HDPE plastics are OK for No Chilling. I am personally content enough with this theory to go ahead and do No Chill, but we should definitely recognize that the science of plastics is still evolving, especially in the area of human toxicity. If it makes you too nervous, you can experiment with No Chilling in metal corny kegs, I've heard of a lot of guys doing that (with mixed results, however).
The lack of a sharp temp drop at the end of the boil means that all your recipes will have to be adjusted. I have experimented with this many times, and am still not convinced of how to accurately replicate 15min-to-flameout additions. I tend to not brew a lot of hoppy styles, so its not an issue for me, but if you are an all-IPA brewer, perhaps No Chill is not for you. YMMV though, as I see a lot of guys claiming to brew really hoppy beers with No Chill all the time, I just haven't personally perfected it yet.
Finally, there is a risk of contamination in a No Chill vessel if (a) the vessel gets compromised somehow (bad lid, leaky seal, etc), or if you waited too long to pour the hot wort into the tank and it failed to 100% pasteurize the vessel. I lost 2 batches to what I believe was a faulty seal in the lid of my main No Chill vessel last year. The first one was just sorta funky at pitching time, the second one swoll the tank up like a tick about 2 days after brewing. It was probably near to rupturing when I dumped it. If your tank swells at all, or if the wort doesn't smell perfect when you open the tank, don't bother fermenting it.
The benefit is that you don't have to chill the wort with an immersion chiller or cold bath. You can set it in the cube and leave it to reach pitching temp. The downside is that you have another vessel to clean and some people worry about the effect of not chilling your wort. Most people like the effect of cold break, but this is a new method so it's bound to be controversial.
I can see you've asked a few questions on this topic. As far as how long can you leave it, I wouldn't go past pitching the next day. The cleanliness of your equipment may vary, and every day you don't pitch is another day another organism can colonize your batch. For materials, the Australians say the use Jerry cans. I think you need a material that can withstand the heat and not leach into the beer. I'm sure a simple search will yield something.