Celebrim
Legend
Would you characterise either character as a single digit level D&D character?
No, but neither would I characterize 1st level D&D characters as being the sort that can survive 200 foot falls.
If I want to play Batman, starting from 1st level, what level would I need to be in order to actually be Batman and how long should it take (as in how many sessions) should I wait until I can legitimately consider myself a reasonable representation of Batman?
If you sincerely want to play Batman, starting from 1st level, then you can do it from 1st level. You start play as the boy Bruce Wayne, orphaned by the murder of his parents and consumed with a thirst for vengeance against all criminals - but still at the same time infused with the idealism of his father. That is a character concept.
And you become the Batman in play. After all, Batman wasn't always the high level hero, the peer of Superman, member of the Justice League. He started out as a crime fighter without a cape, seeking out esoteric wisdom from those masters who knew its secrets.
Is a 1st level character equivalent to Batman? 5th? 10th? Using D&D rules, how long is a reasonable wait until I can claim the cowl?
Batman was a 1st level character once. And he was a 5th level character once. And a 10th. Certainly by the time you are 10th level, you are a superhero and so long as most of the world is merely human, you'll be able to have super-heroic adventures. I honestly think when he conceived of the cowl and took off the ski-mask, he was 7th or 8th level. Although really characters from comic books don't have levels - they all simply have the power of plot. Few superheroes show that better than The Bat.
I think your question misses the point entirely. Batman is a highly experienced adventurer. By the time he returns to Gotham, finds the Bat Cave, and dons the cowl he's already had many adventures and established all sorts of relationships with the setting that will return to haunt him - reoccurring villains and lost loves. To play at 1st level is to start fresh, just beyond childhood or the things of childhood perhaps, at the beginning of a character's career. To start at the Batman, whatever that means to you, is to start in the middle or even past the middle. You start as Batman, and you'll never have an alternate identity to pretend to be. But ok, that's alright. You can start in the middle - in media res - which for you seems to be the start of ones career as a superhero - say 8th or 10th level. But if you want to do that, you can't complain that you can't start as a 1st level character, because 1st level characters aren't meant to be superheroes. You can't complain then that you aren't able to be a superhero by default in D&D at 1st level, because 1st level was never meant to be the start of a superhero career. If you really want to start as a superhero by default, you must play a game where starting at the beginning of a superhero's career is the default - say Mutants & Masterminds or even in fantasy form Exalted. The story D&D tells is the story of becoming a superhero, and then perhaps something more if you continue play - something of a legend or a demigod.
So, again, Batman was a 1st level character once - a young emotionally wounded man named Bruce Wayne. You can play that at 1st level, and that's where I start and that's the sort of characters I create - generally with no idea at all of what their story is going to be because I won't be the only one making the story. If you want to play Bruce Wayne later in his journey, you'll need to start play at higher level. If you want verisimilitude to the superhero genera, you are probably better off with systems specialized to producing that. I recommend Mutants & Mastermind as a comic book simulator, though if your idea of The Batman is the peer of Superman, you can't start at the default PL10 either. GURPS has its charms, though you can't make 'The Batman' as a 100pt character either, you could make Bruce at some point prior to his return to Gotham - though you'd have a longer road to superhero perhaps than you would if you played D&D! Like D&D you'd start with a Superhero's allotment of character points - say 300 - if you wanted to start as a superhero.
In so many threads, you've evidenced a desire to start at the end of the story because you've never before gotten there. I suggest you just do that.