First, nobody “predicted” anything. Models make projections based on the initial conditions at the start of the model run. Model runs with CO2 levels progressing in the same way as CO2 emissions have progressed in real life project warming similar to what we’ve seen in real life.
Here’s an example of a very early climate model, from 1981: http://www.realclimate.org/index.php/archives/2012/04/evaluating-a-1981-temperature-projection/ Note that their projection of warming was 30% below the real life warming.
More model to observations here: http://www.realclimate.org/index.php/climate-model-projections-compared-to-observations/
The urban heat island effect that you describe is well known, and raw temperature data is adjusted to factor out the urban heating effect.
You are misinformed about the state of the science re the Medieval Climate Anomaly and the so-called Little Ice Age,