There are many types you could use but the one that comes to mind first is Microchip's MCP601 which is inexpensive, works from 2.7V to 6V single supply and has rail-to-rail output swing.
To much heat from the iron. Maybe a short somewhere.
I would not worry if this happened to only one. If it happens to the next one then you should start to worry. Parts fail, sometimes they are dead on arrival.
Your schematic should work but before building it you should determine how much gain you need, X10 is probably too much. Remember the maximum voltage the ADC can measure is 5V so only use as much gain as necessary to make the LM35 output up to 5V at the highest temperature you need to measure. Using too much gain will limit the highest temperature you can measure and also make it more prone to amplifying interference.
If it goes crazy there's one rational step you must take reprogram the chip. it may be caused by wrong connections! e.g source connected to source instead of sink.