I am not saying that you are cherry-picking any ponts. It is possible for anyone to cherry pick their starting (and end) points to get any result.
The proper thing to do is to select a long range such as 1940 - present as just as you suggest.
The improper thing to do is to assume that the trend is linear. The graph you cited definitely is not. It starts out almost flat and is steepest in recent decades (just eyeballing).
I also went back 40 years - they were examples. Pick any range you like, do a proper (non-linear fit) and project it into the future. If you do a linear fit then the best that you can hope for is a minimum value for the trend.
15 years is too small IMO. Climate is about changes over decades which is why my first example was 20 years.
http://www.newscientist.com/data/ima...1639-2_808.jpg
There is no real flat period from 1940-1980 - it decreases slightly from 1940 and then increases to be about the same as 1940 at 1980. So we could change the start and end point a bit and get any trend we want.
Once again you are back to using too small scales in your analysis . The years from 2002 are definitely "statistically insignificant" as far as climate is concerned. The problem with using from 1995 onwards is that includes the 1998 peak (caused if I remember correctly by El Nino?) and 15 years is IMO too small.
In any case the climate is not a linear system. It is a complex system with drivers (e.g. CO2), cycles (e.g. solar variations, El Nino), feedback, etc. Periods that increase slightly, decrease slightly and are flat are to be expected.
It is not hand-waving to say that a graph that looks to the unaided eye more like a power curve should not be fitted with a straight line.
I do. I am sure that my greatgrandfather said that winters were colder then

!
That we are still around has nothing to do with the climate. Global warming will not make the human race extinct.
That our population has expanded has nothing to do with GW - even if the worse happens it will have only a tiny effect on the global population.
That longevity has increased has nothing to do with GW, etc.
Some of the negative effects of global warming (of even 0.5 C IMO) fall into the category of the straw that broke the camel's back.
For example the IPCC commited commited a real blunder in citing the WWF report on the Amazon forest. They should have cited the papers that the report was based on (including some that the WWF foolishly did not cite).
What the IPCC and peer-reviewed science say about Amazonian forests
The last paper also emphasises the role of feedback in creating more climate change.