1. Most companies will cover you through the last day of the month in which you serve out your final notice. So if you give notice on December 15, that means they don't have to cover you for longer than the month of December anyway.
If you give them notice that your last day would end up say January 6 however, most of the time you'd be covered for the entire month of January. So don't rob yourself of an extra month's subsidized coverage if possible.
2. You do not have to do COBRA - no one can force you to do it, but a few things about it: you obviously know you'd pay full price (the company stops kicking in the lion's share) so way more expensive. But COBRA is also retroactive; you can wait to decide if you NEED coverage and go in and pay for it to have you covered from your last day of company coverage even a month after the fact. I believe it is 60 days from the last day worked, but you'll need to confirm this. And if you do decide to activate it, it will need ALL the previous time paid for before the 60th day to be valid. So you could wait on coverage, end up needing a doc visit or have an emergency visit two weeks after you left company coverage, activate and pay in full on the COBRA and then have all billers resubmit their bills to the now active COBRA.
Also, if you do COBRA for a few months, you can't get the subsidies from the ACA until the next open enrollment period as you won't have a qualifying event to get coverage through the ACA. You have to either pay full COBRA, or pay full cost on ACA, or get insurance through a broker in the open market - no subsides or cost sharing since they deem you able to get coverage through COBRA or other means. So you'd need to decide how much different and good the savings and coverages are for what you'd be losing out on if you forego ACA right off the bat.
Again, this is how I understood it to work, so do your own confirming before trusting some stranger on the internets. ;)
3. You can have no insurance at all or really cheap but low coverages now since they've killed the penalty for not having coverage/minimum standards for coverage. It used to be you could go 3 months before they'd assess a penalty, but as it stands, you could just self insure, get some very basic "for emergency" coverage, OR get ACA coverage (even bronze level can be good).
4. If you miss the window for open enrollment, then your coverages on ACA would need to be enrolled/paid up by the 15th of month 1 to get coverage on the first day of month 2. So you can't apply on say the 17th of month 1, and expect coverage to begin in month 2; that would occur on month 3.
5. As far as documents - all you need usually is to tell them you retired/no longer are working/earning a paycheck. I think I had to send an explanation letter the first year due to the drop in reported income and no tax return yet to prove, but all I did was create a word doc/pdf stating "retired from work, income derived from investments/retirement accounts, estimate yearly income to be roughly $XX,XXX in 2016." and that was all they needed to approve. I have had no issues in the time after that renewing my insurance on the ACA with estimating my salary as it's always been easy to monitor and control the amounts that count towards reportable income.
EXTRA INFO: keep in mind that most companies pay their matching/bonuses in the first month or two of the new year, so if you seperate before they pay into your retirement accounts or before any earned bonuses are in your account, they don't have to follow through. Some may have it stated in their employee paperwork, but in general, things like matches/bonuses are for RETENTION, not folks that are leaving and unless they are a really, really generous company or it is explicitly stated in legally enforceable text you can take to court, they don't have to pay you for that stuff. If it is a smaller business, I would make sure to get all that is coming to you before giving ANY notice.