From old sources, it says "the first 16k of PRG ROM at $8000 and the last 16k of PRG ROM at $C000", but is this correct for most of the mappers?
It depends entirely on the mapper. I think Disch's documents on mappers have relevant mapper powerup warnings for the mappers with games that depend on powerup values.
Not necessarily. Some mappers have a well defined start up state. Others do not. To make matters worse many iNES mapper numbers are assigned to several chips that vary in how they behave. So some mapper X games expect a well defined boot state, while others do not. Often the older revisions of a given mapper have an undefined power on state and a later revision will add one. (The MMC1 is the most prominent example.)
In general, in these ambiguous cases, one can handle it by applying the old adage "Be conservative in what you generate. Be generous in what you accept." For an EMU dev, this means that mappers should start out in a well defined state, so that you don't break games that where written to be deployed with a hardware revision that has one. For homebrew dev, this means you shouldn't assume a well defined startup state if there's the possibility of being deployed on a harware revision without one.
Disch's mapper docs are usually pretty good with regards to covering this stuff. In particular, he gives separate advice for emu and homebrew dev when pertinent.
MMC1 is MMC1=Random, MMC1A has non-random with no WRAM disable I think then MMC1B2 I know always boots in bank F. But don't take my word, check the docs. I check kevs for MMC1 stuff for development like that.