Hell we have hard enough time predicting the weather beyond 5 days in the future.
I think I see why you have trouble understanding the basic science, weather is NOT climate. Climate is the statistics of weather, unlike weather which is mathematically chaotic and is currently impossible to predict more than a few days ahead, climate is extremely stable over human time scales, climate forcings [wikipedia.org] are relatively well understood and lend themselves to finite element analysis in the same way as many other complex natural phenomena are modeled for scientific and engineering purposes.
What leads me to trust these predictions 50 years from now?
A better question to ask is what leads you to doubt "the republic of science" has failed to lead to the best available answer on this particular question, emotion, ignorance, corporate propaganda, intellectual laziness? - There's certainly very little logic in your post and even then is based on a common (and fundamental) misunderstanding that psuedo-skeptics have been exploiting for decades.
One thing you will never hear these scientists say is, "We predict the Earth will warm but honestly we just don't know, it could end up getting cooler."
You hear it all the time, you are just not listening, uncertainty is expresses with something called error bars (as seen in the WP link above), you should learn how to read them because they are quite useful from a risk management POV. For example, the insurance industry has been routinely offsetting the risks identified by climate scientists for the last 10yrs by adding the expected costs to your premiums.
As for TFA, the "4 degrees by 2100" prediction has been widely accepted by climate scientists for over 20yrs and is based on a "business as usual" scenario. The only thing about the prediction that has significantly changed is the certainty of the prediction (ie: the error bars). The scientific advise is to try and limit the increase to 2 degrees to avoid further unnecessary death and destruction.
Replacing the planets energy infrastructure may seem like a herculean task, but my bet is it will happen right under our noses and when it's done people will forget how far we have come (as they have already done with acid rain and pea-soupers). Not a single coal fired plant currently operating on the planet existed when I was born (1959), every one of them has been built (and often rebuilt) in my life-time and they are now much cleaner, but they now also service more people than there were on the entire planet when I was born. I believe science and common sense will prevail and we will adapt our infrastructure rapidly over the next 30-40yrs (the working life of a coal fired generator). Luddite billionaires who continue to deny reality will be left sitting on a worthless coal mines bombarded by negligence lawsuits from anyone with so much as a wheeze.