Why make up a new background? Plenty of the existing ones assume that you're experienced and have been in that role for some time.
It's possible to write anything. Writing it though doesn't make it coherent.
Criminal states in its first sentence that you're experienced and have a history of breaking the law, and portrays you as a well connected member of the criminal underworld.
Having a history of breaking the law and being a well-connected member of the underworld doesn't necessarily mean you aren't 1st level. But that you would be experienced at 1st level is basically impossible and incoherent to setting. To begin with, if you have been engaged in criminal undertakings, then these are inherently dangerous pursuits. If you were to engage in them as a criminal in game, you would gain XP and begin to level up. After you'd been doing it for a while, you'd no longer be 1st level. It therefore is obvious that you couldn't have been doing such things for very long, just however long it takes to go from 0th level to 1st level (which itself is probably less than it takes to go from 1st to 2nd). Secondly, if we were to detail the criminal underworld you are a part of, producing a detailed hierarchy of a Thief's Guild (perhaps modelled off the gritty crime family in The Soprano's) it would be obvious that as a 1st level character you were one of the least experienced members of the guild. If we used the exceptional 2e Thief's Handbook to generate the guild, this would be obvious. You might be well connected - you might be "Tony's" nephew - but you'd clearly be one of the least experienced - maybe not least talented by certainly least experience - members of the group.
Folk Hero explicity gives you a heroic past.
The same problem occurs here. If you have been a hero then the things you have been doing if played out in the game are worthy of XP reward. So at most you might have done one or two small things that garnered you some reknown and now you are most famous for being famous, but you can't possibly be an experienced hero and be 1st level and that be coherent.
On the other hand:
Guild Artisan specifies that you've risen to being both well-established within the mercantile world and a master of your craft.
As a Hermit you've lived in seclusion long enough to have made some remarkable philosophical discovery.
A Sage, like the Guild Artisan, has spent long enough in their profession to achieve mastery, this time in a particular field of scholarly study.
Those three backgrounds involve backgrounds which present no real contradiction. They are all ordinary professional backgrounds. The game rules generally contain no provisions for gaining XP for doing ordinary things that aren't dangerous and don't present risk. So you can totally be an elderly hermit who has never faced anything but ordinary hardship and it make sense that you are 1st level, because if we played out your backstory it would be boring and you'd gain no XP. Again, if you want to produce a backstory coherent to the assumptions of the game, you need to produce one that explains why you haven't faced a lot of risk and extraordinary hardship to this point.
Incidentally though, nothing in the 5e rules actually supports the claims made by the backgrounds. Regardless of whether you claim the Sage has masterly knowledge of a field of scholarly study or not, in actual practice they have no mechanically abilities that prove that and you will suffer in play compared to any NPC scholar that actually has those abilities in game.
Soldier carries with it a sufficient military rank to command the respect and deference of others, not something easily achieved without a significant career.
You are thinking about this from a modern perspective with military meritocracies (at least in theory). Officers in the historical eras that the game is drawn from frequently held their rank entirely for political reasons, by virtue of birth, or by virtue of having paid the crown to be promoted to that rank. And while social status is not a particularly significant thing in modern times, it was a big deal at one point. It's quite easy to imagine a 12 year old with sufficient military rank to command the respect and deference of common soldiers in a historical setting.