do the ninja legally exist?
Wu doesn't legally exist. He and Garmadon predate all modern governments. Garmadon only exists on two forms: his marriage certificate and lloyd's birth certificate as the father.
Lloyd, as mentioned above, has a birth certificate, all the stuff you get when you are born like an ssn, and he has school records from his time at darkley's. they are VERY out of date and is the reason he does not have a driver's license, or any form of ID. he doesn't want to have to talk to a government employee about Magic Tea.
Kai and Nya probably have information from their births, but it's lost somewhere in their family's attic and both were abandoned too young to know/care about that kinda stuff until they're much older and have an "oh shit" moment when trying to get a driver's license.
Jay's information is safely filed away, and his parents badgered him into a learner's permit before the series even began. sadly, however, the series started his wild adventures, and he doesn't realize until much later that he never a) got an official driver's license or b) updated his permit, so it is no longer valid.
Cole's dad didn't let him leave the village to go to "school" without a driver's license. Not just a permit,but a full license. Cole was good at keeping it updated, but life got out of hand and he ended up forgetting.
Zane doesn't legally exist and decides he doesn't want to explain to a DMV worker why that is, so he just hacks the DMV and prints his and everyone else a driver's license, as well as periodically updating them with an automatic program.
Pixal and Zane actually do have a conversation later on about nindriod legal rights and decide to bring it up to the government. it's not as big an issue as they thought, considering nobody who would be in charge of setting up IDs would openly discriminate against two public heroes, even if they had those beliefs.
also i should note this is mostly centered around early seasons. they would sort this kinda stuff much later. possibly -at the earliest- post the great devourer battle, when they have to live in ninjago and rent apartments. another chance would be after the final battle, when wu wants to run a school and they somehow become teachers.