To the original questions:
In 5e at least, story and background trumps hard rules in terms of where your character comes from and what kind of training they have. It's left predominately in the hands of you and your DM to decide what works ad makes sense for you. You might play a middle-aged and fairly grizzled former watchman who is striking out on adventure or is dedicating themselves to a greater oath of righteousness now - their history is the explanation for their above-commoner level capabilities and experience. OR, you might be playing a seventeen year old halfling girl who ran away from home to see the world, and has only recently discovered the beginning blooms of the kinds of magical power her music, song and dance can evoke if she puts her heart into it... Your exact amount of experience in an adventuring lifestyle is almost entirely between you and your DM. the game rules give you as et of proficiencies and skills to represent your beginning point as an adventurer, but you can decide with your DM how you came by them, or even if you have access to all of that right away, if you want.
None of this is new to 5th ed. You could play a person with zero experience at anything in any edition if you wanted.
This doesn't change the fact that the game assumes your character has quite a lot of training even before starting his adventuring career. The new Background feature only adds to that assumption by providing even more pre-existing skills (as well as the means by which you explain why you have them).
You can choose to play a smith's apprentice who's never held a weapon before in his life, but he still has the skills of a well trained warrior because you chose Fighter as his class. You can choose to play your halfling Bard who's only just leaving home, but she's still a well trained musician (proficient in three instruments) and already trained in a very specific form of magic. This isn't "background and story trumping hard rules", this is the opposite, this is game mechanics trumping background and story.
