In terms of BTO prices, I've heard some people saying how some areas are more expensive than others. What is the average, standard BTO price?

BTO in matured estates are usually more expensive. BTO at good locations (near present or future amenities) are also more expensive. Price also varies by floor.

The prices are set by hdb. They will usually have an estimate on the hdb site (below is an example). They give an estimate cause the price differs based on the facing, location, height, etc.

https://esales.hdb.gov.sg/bp25/launch/20nov/bto...

All BTO are below market value as you are basically investing both time and money for it to be built as compacted to resale which only requires money (cause it's already completed).

If you want to compare with past BTOs in the area, you can check 3rd party sites like 99.co as they usually sate them