Answers:
1. You can avoid underpayment penalties by meeting any of what are called the "safe harbor" rules. There are at least three different safe harbor rules. One of the safe harbor rules is paying in at least 100% (or 110% if your AGI was above $150K) of your 2018 taxes due, either through withholding from your job or estimated tax payments. Another of the safe harbor rules is paying in at least 90% of your 2019 taxes due, again either through withholding or estimated tax payments.
If you do not meet any of the safe harbor rules, then you must make estimated tax payments to avoid the underpayment penalty.
If you choose to make estimated tax payments, then yes, you would use the payment vouchers at the end of Form 1040-ES.
2. You're correct in that the estimated taxes due to your LTCG would be $9K. The IRS looks at your whole income picture in determining taxes and estimated taxes, so you would want to account for your other income too (job, pension, SS, etc.). If your other sources of taxable income have enough withheld, then yes, you can just pay the $9K.
3. Estimated taxes are due four times per year. The first estimated tax payment due date is April 15. The other dates are listed in Form 1040-ES. So you could pay $9K by April 15 or you could pay $9K/4 = $2,250 on each of the four due dates.
4. You can if you want. Or you can mail a check. Or any of the other methods listed in the instructions for Form 1040-ES.
5. Keep whatever proof of payment you have. Either a copy of the canceled check, or a receipt from Direct Pay, or whatever receipt is given based on your method of payment. Unless you have an unusual situation, though, paying estimated taxes probably won't increase your chance of an audit any appreciable degree AFAIK.