>Lv 1 is challenge run category
Yep. Though some say it's 'the true highest default difficulty', since you're not really locked out of much stuff (mostly passives) if you do this. If it weren't for the fact that devs outright said that they were surprised people did such runs, most people would have assumed that they designed the game around Lv 1 (since everything just works shockingly well, so you might as well say that it was a lightning in a bottle sort of deal).
>any others
Many. Lv 1 is used for most of those. You can go for 'No Magic', 'No Drives' (though the Cavern of Remembrance becomes an exception, not because of combat, it's due to platforming requiring their use), 'No Summons', 'No Items', 'No Limits'... and the most difficult being 'No Abilities'. You can also mix and match these, though I don't remember if you had to draw the line somewhere.
KH1 doesn't work too well for Lv 1 (only added in the HD re-release on PS3 and newer consoles). The main campaign is doable, but as far as I remember, one of the time trials is impossible. Also, unlike in KH2, you don't get abilities (from combo modifiers to the ability to block attacks *that said, you can still parry attacks if your weapon's hitbox clashes with the hitbox of an enemy attack, but this evidently requires much tighter timing*) via progression, you get them via leveling up. KH1's Lv 1 is also bugged in a way that if you know what to do and don't hold back, you can finish some fights faster than if you were at Lv 100 (as nonsensical as that sounds).
>are they difficult?
On your first go, they can be, even if you're not playing on the highest setting (unless you go out of your way to overlevel *which is to say, deliberately backtracking for hours to get loads of experience, but you can still kill the things that stand in your way*). But once you get the hang of how a few moves (and enemy AI) work, things will become notably easier. Many of them still require some sort of agency, though.
Lategame (in both KH1 and KH2), you get two game breaking abilities that... well, break the game in half. But don't worry, they're just passives, so they can be ignored more easily. They're also gotten via leveling, which is mostly why people say that thing about Lv 1 being the best way to play KH2.
>start on hard right away
KH1's Proud (Hard) is the hardest in the series, and at times (read: rare) it's as difficult as Critical (Very Hard, this one was introduced in 2), but it also depends on whether you go out of your way to overlevel. You can learn the ropes on Normal (Standard), then try Hard next (Proud). KH2's Proud is easier, so you can probably start on that one, then do a Critical run next.