You're in a bit of a reverse engineering mode here. You could, for instance, look at the data pin on the DHT11 using and oscilloscope or logic analyzer checking for periodic activity. The expected waveform is described in the DHT11 datasheet. If this happens then there is a decent chance that the processor is running as expected, but that's not a certainty.
I'm guessing that the program wouldn't be spitting out strange characters regardless of the sensor readings, and in particular, it's unlikely that multiple sensors would have failed. Thus it seems likely you have a problem on the display side of the system. It uses a 4 or 8 bit parallel data interface and a broken connection on that bus could plausibly cause the symptoms you're seeing.
If you have another Arduino board available, you might try hooking the display up to that and running some example code from a reliable source. You could do the same with all the peripherals.