If you are not going to have a covid disaster qualified distribution, good for tax year 2020 and for up to $100,000 (making it entirely penalty free), then you just need to make sure you are not mixing up the provisions.

Home buyer provision: "Once you've exhausted your contributions, you can withdraw up to $10,000 of the account’s earnings or money converted from another account—without paying a 10% penalty—for a first-time home purchase.

If it's been fewer than five years since you first contributed to a Roth IRA, you'll owe income tax on the earnings. This rule, though, doesn't apply to any converted funds. But if you’ve had the Roth IRA for at least five years, the withdrawn earnings are both tax- and penalty-free."

So now we see "tax free vs penalty free" or both, or neither. Now you do your analysis.

The/any Roth account needs to exist for at least 5 years. That really meets the same concept of "the money had to be here at least 5 years" in startup mode. What it doesn't meet is the additional 5-year rule (conversions), that "other money put in might have its own 5-year rule." That's why I gave the link to the investopedia article, because they explain there are multiple 5-year rules.

So, to review: if the/any Roth account hasn't existed at least 5 years, then there is No Money in it that type of account that has been there 5 years. By Definition. No matter how the money got there (contribution, conversion, earnings). Hence, earnings distributed are taxable, even if for Home Buying, Education, etc.

"I do not think there is an exception to the 5 year rule, at least I cannot find from the official record. There is exception for 10% penalty only."

The exception is not also Exclusion from income tax on the earnings, but Conditionally.

Consider Conversions: subject to Income Tax, but excluded from additional 10% early distribution penalty at the time of conversion; not subject to income tax again, even if distributed early. Subject to penalty if distributed early.

Earnings distributed for First Time Homebuyer is possibly excluded from both tax and penalty (it's subject to the 5-year account existence rule, as noted above, for the income tax part). Also, from a different investopedia article: "The Internal Revenue Service (IRS) defines that status rather loosely. You are considered a first-timer if you (and your spouse, if you have one) haven't owned a home at any point during the last two years."

https://www.investopedia.com/articles/personal-finance/110415/can-you-use-your-ira-buy-house.asp

In the investopedia articles, the IRS references are links given inline and at the bottom:

https://www.irs.gov/taxtopics/tc557

"Exceptions to the additional 10% tax apply for early distributions that are:

• Made to a beneficiary or estate on account of the IRA owner's death
• Made because you're totally and permanently disabled
• Made as part of a series of substantially equal periodic payments for your life (or life expectancy) or the joint lives (or joint life expectancies) of you and your designated beneficiary
• Qualified first-time homebuyer distributions
• Not in excess of your qualified higher education expenses
• Not in excess of certain medical insurance premiums paid while unemployed
• Not in excess of your unreimbursed medical expenses that are more than a certain percentage of your adjusted gross income
• Due to an IRS levy of the IRA under section 6331 of the Code
• A qualified reservist distribution
• Excepted from the additional income tax by federal legislation relating to certain emergencies and disasters (see the Instructions for Form 5329 for more information), or
• Not in excess of $5,000 and the distribution is a qualified birth or adoption distribution (see the Instructions for Form 5329 for more information)"

These numbers you want us to "do the math" are different than your first taxpayer's?

Let's state it as:

Direct contributions distributed are not subject to tax or penalty.

Any conversion has a 5-year clock, so anything put into that account from 2016 and later, removed in 2020, is Early. That = penalty (not also taxes), unless you want to show this is qualified under one of the penalty exceptions.

Earnings are taxable if early, if not having the/any Roth for 5 years, when not used for the tax exclusion purpose: Home Buying.

https://www.irs.gov/publications/p590b

I think that covers how you need to break out the funds, compare Basis to the Distribution Ordering, watch for 5-year spans, and see what comes out the other end. I've done my best to review all of this. Let me know if you see any errors.