"if you don't claim in a certain amount of time, like 90 or 160 days, I forget, 'they' choose the 30 installment plan for you. "
The IRS gives 60 days to make the decision, but the clock only starts when you claim the prize. IIRC it goes back to the switch from annuities as the only choice to the option to chose between the annuity or the lump sum, and is based on the doctrine of constructive receipt or economic benefit. Just because somebody promises/owes you money doesn't mean that you'll get it, so the mere promise of money doesn't make it taxable. Once the lottery is actually ready and willing to give you the entire (lump) sum it becomes taxable income. It's only be restricting your access with an annuity that constructive receipt of most of the prize is deferred, and therefore not taxable in the year you get the first check. That's why you can't change from lump sum to annuity once you've selected the lump sum. By selecting the lump sum you've taken constructive receipt and owe taxes on the full amount, with no option to wait on collecting and paying taxes.
"instead of giving family & friends money. I'd loan it to them." "my idea probably is either not legal or will still result in me paying some type of tax"
It's legal to loan money to friends and family and it's legal to make annual gifts so I think the overall scheme would be legal, but somebody will pay taxes. You can loan them the money, but of course you have to charge interest at a fair rate or it's considered a gift. Let's pretend that 2.3% is realistic. If you loan somebody $1 million for 30 years at 2.3% the monthly payments will be $3848, which comes to $46,176 per year. In the first year the interest portion is very nearly $23,000, and over the life of the loan you'll collect $385k in taxable interest income. If you've won the lottery and have enough to give somebody a million bucks you're likely to (currently) pay 37% in federal taxes on that income. That's about 142k, but you get to spread it over 30 years. Since it's front end loaded let's just call it an even 10% effective gift tax, which is a lot better than the usual 40% tax rate on gifts. The recipient would have to pay income tax on any income from the money, but that would be pretty insignificant compared to the gift tax rate, and they'd still be paying that income tax even if the money was an outright gift.
You've also got to make annual gifts of $46,176 so they can make the loan payments, but that money comes back over the next 12 months, so it's a wash. If it turns out to be unlawful this is where/why I expect that would happen. By making the annual gifts you're effectively waiving the required repayment. Waiving the repayment isn't illegal, but it does have the unpleasant tax consequence of turning the $1 million into taxable income for the recipient. Because the AFR rate is lower than a fair market rate making the loan and then making annual gifts of enough to make the loan payments could easily be viewed as a tax dodge, and that certainly seems to be the intent as far as this discussion goes. Figuring out if the IRS has specifically addressed this scenario requires more than a quick Google search.
Circling back to constructive receipt and economic benefit, I think there should be a giant loophole. The IRS is on record, as a result of a legal case, as saying that you don't have constructive receipt and the resulting tax liability until sometime after submitting your claim. I view that as meaning that your ticket with the winning numbers is still only worth its face value until you submit the claim. If that's the case giving somebody a 1% share in the ticket would mean you've given them a gift worth a few cents to half a buck, depending on the cost of the ticket. That's well within the annual exclusion, so it wouldn't trigger any gift tax. OTOH, since that case the IRS has issued rulings that sharing a ticket does constitute a sizable gift, but I don't know if the winners in those cases argued the economic benefit angle.
Finally, you can always have an agreement to share any winnings even before buying a ticket, and I don't see any reason that the shares can't be very unequal. You'd just have to make sure that the sharing agreement is in writing, and can be enforced by all parties. Of course it would have to be for a portion of the ticket(s) rather than a set dollar amount, and if you won 10k or $1 million you'd have to share the prize even though your real intent was to avoid gift taxes on the gifts you could afford if you won a big jackpot.