The first is experimental design, much like is necessary in any aspect of science: Annual tree growth is definitely a response to a variety of regional, site-specific, and tree-specific factors for any given species in a location. So in order to increase the 'signal to noise' ratio for a given signal (temperature, precip, fire, disease or whatever reconstructions), you look for combinations of species and sites for which the factor you want to study is the limiting growth factor; in other words, other things the tree needs to grow are present in abundance and one is in shortage. If you want to make a precipitation reconstruction, you find a site that has very little soil moisture storage, such as trees basically growing out of fractures in steep bedrock. If you want to study temperature, you pick species like Bristlecone Pine that grow at high elevations (which are typically relatively wet), so that the temperature/length of growing season is the annual limiting growth factor. That sort of thing.
The second is that researchers use many other forms of measurements than simply ring width to tease these problems out. For example, many people use oxygen isotopic ratios[2], latewood (the dark outer part of the tree ring) density measurements[3], and other measurements in tree rings, both because of the unique and accessible response of the tree, and because (unlike, say, sediment cores) the tree rings are much more quickly, accurately and reliably dated at the single-year precision.
Third, there is a lot of statistical analysis thrown at the problem. I didn't work a lot with this, or see people in my lab do it, but I am aware of researchers doing multivariate statistics techniques such as principal component analysis[4]. I'm sure there is a lot more, but you can do a Scholar search as easily as I can.
Fourth, one of the strengths of the multiproxy technique is that many of them are independent, and they all show a lot of correlation to the temperature records during the instrumental calibration period. There is some divergence before this period, but not as much as would be expected otherwise. There is also the problem of weighting, like you mention, and I don't think anyone is comfortable with it but I don't think it's as ad-hoc as you make it out to be. A review (by a statistician) that discusses this is here[5]; I didn't read it in depth.
[1] http://www.uark.edu/misc/dendro/
[2] http://link.springer.com/article/10.1007/s00382-013-1674-3
[3] http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-11-00139....
[4] http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0706.2010.... [5] http://www.stat.unc.edu/postscript/rs/Sensitivities-Submitte...
True, the signals from tree rings (etc.) are noisier than those from kitchen thermometers. Let's consider, for a moment, how you could verify the precision and come to a conclusion as to the accuracy of those signals: well, when you get the same information coming from multiple seemingly independent sources, it greatly increases your confidence in that information. Hence, if you're doing your modeling right, if you get tree ring data from all over the world, and soil samples from all over the world, and find that, sure enough, there is a common signal visible across all of them, then you say "well, the only thing that can explain this global correlation is the modulation of some global parameters". And you present your data and conclusions to a bunch of people whose job it is to poke holes in it and prove you wrong, and after a bunch of time eventually people whose job it is to prove each other wrong end up roughly agreeing on a small sliver of information that lies at the intersection of all their understandings of the world.
Signals in physics can be noisy too (fortunately, in physics you can take millions of samples). But to say "well, these error bars are too big, so therefore we shouldn't pay any attention to the trend at all -- we can decide to do something when we have much higher confidence in the precise figures. In the meantime, I'm going to oppose tighter fuel economy standards for vehicles, and revenue-neutral carbon taxation, and aggressive subsidization of alternative energy R+D, ... " because god forbid we should all drive around in smaller cars, or that we should stop subsidizing air travel, or that we should too quickly improve solar cell efficiency.
Excluding math, we use proxies for just about all science. With different assumptions and different criteria. I don't deny that there are politics, assumptions and margins of error but they do put effort in to minimizing them and doing real science. Take a look at pharmaceuticals, they don't even know why a lot of medicine works, in fact there is debate on if it even does... But people still buy it and take it.
Proxy temperatures, statistical methods, atmospheric modeling, etc, are all scientific areas whose knowledge can be gleamed at sufficiently well for a person with some form of higher education in science or math education (but only barely; remember that this stuff is being researched by people who have devoted their entire lives to the scientific specialty). But that is an astoundingly low proportion of the population. And even then it is clear that, as methodological errors have diminished and methods for more accurate models have appeared, that they all tend towards the same conclusion.
We should be worrying significantly more about the small but non-trivial (and ever-increasing) probability of an extinction event within the next decades rather than pretend that methodological errors, at this point, are a significant issue in identifying the climate trends we have spotted in the last decades.
There's a billion pieces of supporting evidences for global warming. As far as I have ever heard, there's not a single piece of evidence against it, just piles and piles of FUD thrown at each model that tries to explain it.
Surprisingly enough, the fear, uncertainty, and doubt about anthropogenic warming claims have come from people who receive money from carbon-intensive industry groups. It's these people who keep on claiming that it "hasn't been accomplished". And since FUD is easy to do, the FUD-purveyors are winning in the public sphere.
What kind of evidence would convince you, exactly?
I.e. maybe the climate won't spiral out of control, but it seems reasonable to be concerned about the alarming levels of CO2. As a new parent I'd love to have my dire fears of catastrophe awaiting future generations assuaged by rational hard science, but (as a layman) everywhere you look the picture is pretty bleak.
The main problem is that no one knows if the hypothesized models (temperature vs. tree ring thickness, sediment layer thickness, etc.) have any validity. Why should tree rings grow linearly with temperature- what about a huge number of other variables? The temperature “signal” is at best very weak. Decoupling the temperature signal from other variables (humidity, elevation, tree species, CO2 concentration, etc.) is a hugely difficult multivariate statistics problem and most of the researchers I've corresponded with do not seem to appreciate this fact.
Another major problem is that proxies get weighted in the temperature reconstruction based on how well they fit the hypothesized temperature model. The weighting is where things get really shaky. Right away, you have biased your results in such a way that that nonconforming data is discarded / de-weighted. Note that there is not anything wrong with these nonconforming samples that invalidates them; they just don't fit the proposed models very well. Typically researchers develop their own methods for weighting samples in the temperature reconstructions and many of these methods would make a statistician cringe. Often during the weighting phase you find that a handful of proxy samples (out of a sample size of hundreds) get the vast majority of the weight in the temperature reconstructions. Sometimes the weight of a temperature reconstruction in a location is almost completely computed from a few proxy samples taken halfway around the world at a completely different elevation.
If you try to discuss these fundamental problems with climate scientists you'll typically get told that you're not a scientist (not true) or that you're a "denialist" being paid by big oil (not true). I am convinced that these temperature reconstructions are useless at best.