Congrats to your daughter on her success!
As to your questions, here's my take:
1. Her age doesn't matter.
But she could consider it hobby income this year since that's how you describe it. If she reports it as hobby income, it would go on Schedule 1 line 8 under Other income. If your tax software allows it, you would want to mark it as earned income because it clearly is.
The other option is to consider it self employment income. This would be reported on Schedule C. She does not need a business structure, business name, or business location to file a Schedule C - she can be a sole proprietor under her own name, SSN, and personal address. You will need to pick a six digit "business code", but just search the business codes for something that seems close - there'll be one in there for artist or something.
If she stops doing it after this year, then hobby income this year makes sense to me and is simpler to report. If she continues next year and the business grows and she gets more serious, then self employment income makes more sense. You could do hobby income this year then switch to self employment income in subsequent years, or you could just start out with self employment income this year to be consistent - I don't think it matters.
2. Yes, she can open a Roth IRA without filing a tax return. The $400 limit has to do with the fact that self employment income over $400 does incur self employment taxes, so if she makes more than $400 per year in net income from self employment, she would be required to file a tax return and she would have to pay self employment taxes. The self employment taxes associated with a Schedule C are calculated on Schedule SE, which would also be filed with her tax return.
Even though she is not required to file a tax return in this situation, some people think it is a good idea to file a tax return showing the income in order to substantiate the Roth contribution. Personally I don't think it is necessary as long as you keep the income records as you have to substantiate her Roth contribution. But filing an optional return is reasonable and permissible if you wish to do so.
3. You don't need a business structure. She would need to file a tax return with a Schedule C and Schedule SE and pay the self employment taxes due. See Chart C in the IRS Form 1040 instructions under "Who must file?"
4. That sounds sufficient to me. If you wanted to get more serious and possibly save a bit on taxes, as a self-employed individual she would be able to subtract her business expenses from her income. Things like paint and canvases and paintbrushes and easels and painting smocks and business cards, and anything else that is ordinary and necessary for her business could be subtracted. This would reduce her net income, which would reduce her SE taxes, and if it reduced her net income below $400, would mean she would not be required to file a return.
5. MDM is correct that the FAFSA doesn't include retirement assets (either yours or your child's). But not all schools are FAFSA schools - some use something called CSS Profile, which may look at retirement accounts, I don't recall offhand. If your kid is going to your state university, then FAFSA is probably the one to look at; if she's going to Oberlin or the like, take a look at CSS Profile.
Vanguard was not very accommodating when I wanted to open custodial Roth IRAs for my three kids about five years ago. They had account minimums that were in the $1K range, and the paperwork was a challenge IIRC. Maybe they're better now.
My kids ended up at Schwab, which had $0 account minimums, were happy to open custodial Roth IRAs, and there was a local office which we went to to open the accounts. We've all been pleased with how things have gone there. Can't speak to Fidelity for custodial Roth IRAs, but I've had other accounts with them (currently my HSA and a Fidelity credit card) and they're very good.
