Welcome, Hcxc1runner
You sure know to ask the right questions. I assume that you have only recently been confronted with the global warming hype for the first time. A few hundred specialists are working on those subject and of course forums answers are biased with the opinion of the answerer.
Let me give you a few tips. Do an advanced search on this forum, with search words "carbon" for any date in Earth science and you'll have some stuff to read. I hope you don't mind that I'm brief on your questions:
Is there data that points to a statistically significant increase in carbon dioxide levels, and temperature increase? Is there data that shows a relationship between the two variables (CO2 levels and temperature increase)?
There used to be a "hockeystick" with a r2 of 99+% between the two, however it has been demonstrated that is was a very ..errm.. *unscientific* graph here:.
http://www.grida.no/climate/ipcc_tar/wg1/005.htm (fig 1 b)
There are several discussions about "noble cause corruption". Was it cheating or just bad science?
Glacial ice cores show a clear correlation between CO2 and isotopes, erroneousle believed to be depicting temperature, however the *temperature* is leading by 600 +/- 400 years during the last deglaciation. Global warmers assume a positive feedback, boosting the temperatures but obviously those have not studied the physics of positive feedback loops.
Finally, we have chemical CO2 measurement as of about 1816 AD up to about 1961 AD with wild oscillations which are not accepted by the IPCC and warmers, however with indeed some correlation. For the last 60 years we have anti correlation between ~1950-~1975AD, cooling temperatures with rising CO2 when the return to the ice age hype was at the top and then a positive correlation between ~1980 and 1998
How are temperature levels measured, and what exactly are they measuring?
There are three independent research facilities working on the compilation of all the world meterological stations and ship meteo data. Average daily and monthly temperatures are calculated per grid of some 5 degrees lattitude and longitude, I think.
See for instance Hansens lab:
http://data.giss.nasa.gov/gistemp/station_data/
Then there is the monthly satellite measurements for the lower troposphere:
http://www.nsstc.uah.edu/data/msu/t2lt/tltglhmam_5.2
How precise and accurate are these methods?
That's a can of worms with big discussions everywhere. Two hot topics are correction for the Urban Heat Island effect and the reducing number of rural stations.
How are sea levels measured?
In the old days with the yardsticks in the harbours, corrected for geologic movements. Nowadays we have the satellites.
Are carbon dioxide levels constant in the atmosphere?
Definitely not, there is an annual wobble due to seasonal changes in sources and sinks, according to the ice cores, CO2 levels have been fluctuated between some 180 and 280 parts per million in the ice ages. However, it is aknowlegded that there are plenty of complications with the CO2 in the ice cores. Anther "proxy" are fossile leave stomata of certain species which are know to show variation in stomata density depending on CO2 levels and those show much more variation and higher levels.
According to the IPCC the CO2 levels started to rise gradually from around 1850 at 280 ppmv to nowadays 380 ppmv on the average. The earlier mentioned chemical measurements showed values between 1000 ppmv and 270 ppmv.
I hope it helps.
Oh well. 
There's a little g in there.