I'd like to learn more about LoRa so I decided to build a few nodes but I'm unsure whether to use 433 or 868 MHz. TTN uses 868 MHz due to the excessive man-made noise in the 433MHz band due to alarms, cars, weather stations, etc.
Can I use an inexpensive 433MHz receiver (pic) to evaluate the noise level in my local area? If so, do I need to amplify the noise coming from the receiver with an OpAmp?
I believe you are confusing the concept of a noise floor and co-channel interference. The noise floor at UHF frequencies is largely driven by the receiver rather than environmental noise so there isn't much point in evaluating it with other than the objective receiver. Co-channel interference is real signals in the RF environment that are received in addition to the signal of interest. Since it is a function of the external environment any suitable receiver may be used to evaluate it.
LoRa is a spread spectrum signal, which is to say it has a much wider channel bandwidth than conventional receivers. Thus one would need a wide band and/or tuneable receiver to look at the LoRa signal channel. It's worth noting that spread spectrum radios are able to operate in the presence of significant levels of interference, which is their principle advantage over conventional radios.
At the hobbyist level a common approach is to use an RTL-SDR receiver with some control interface software such as GQRX or SDR-Sharp to look at the radio spectrum at the frequency of interest. This setup can be used effectively as a radio frequency spectrum analyzer. There have been projects using this sort of equipment to examine and decode the LoRa Waveform.