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You have several questions--but there is one that you should have asked and did not ask.   If you are hoping to get the child care credit on your own tax return, you cannot get that credit for paying your own dependent to provide the child care.   So if your teenage child is the only one being paid to provide care for your younger child, while you work, you will not be able to get the child care credit on your own tax return;

'1. How do parents determine if the minor "provided more than half his own support for the year";

You say that by the end of 2021 your child's income will be about $7000 in total.   Even if he had double that income for 2021----would that pay for his food, housing and clothes?   It certainly does not sound as if he is providing over half his own support.   He is spending his income on typical "kid stuff."

• What does "financially support another person" mean?

"2. Do parents need to withhold any taxes and issue a W-2?"

Do you want to consider your teenager to be a "Household Employee" ?   Or is he just your child to whom you are providing an allowance in exchange for helping in the household?

https://ttlc.intuit.com/community/tax-topics/help/who-is-considered-a-household-employee/00/26770

Child employed by parents.(p13) IRS family help

Payments for the services of a child under age 18 who works for his or her parent in a trade or business aren't subject to social security and Medicare taxes if the trade or business is a sole proprietorship or a partnership in which each partner is a parent of the child.

 If these payments are for work other than in a trade or business, such as domestic work in the parent's private home, they’re not subject to social security and Medicare taxes until the child reaches age 21.

 Payments for the services of a child under age 21 who works for his or her parent, whether or not in a trade or business, aren't subject to FUTA tax. Payments for the services of a child of any age who works for his or her parent are generally subject to income tax withholding unless the payments are for domestic work in the parent's home, or unless the payments are for work other than in a trade or business and are less than $50 in the quarter or the child isn't regularly employed to do such work.

https://www.irs.gov/taxtopics/tc756

"3. Should he file his own taxes? "       

You say he will receive a W-2 for the after school job.  If tax is withheld as shown in boxes 2 or 17 of the W-2 then he would want to file to seek a refund.

MY DEPENDENT HAD A JOB

If your dependent has a W-2 for his after-school job, summer job, etc. you do not include the information on your own return. You can still claim your child as a dependent on your own return.  He/she can file his own return to seek a  refund of some of his withheld wages (he won’t get back anything for Social Security or Medicare), but MUST indicate on it that he can be claimed as a dependent on someone else’s return.  (Supervise this closely or prepare it for him!)

If your dependent’s earnings were over $400 and were reported on a 1099Misc or 1099NEC then he must file a return and pay self-employment tax for Social Security and Medicare. 

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, or fiancé (etc.) as a dependent on your 2020 taxes as long as they meet the following requirements:

Qualifying child

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
• No age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

Qualifying relative

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $4,300 in 2020.
• You provided more than half of their financial support.

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them. There may be other tax benefits you can get when you claim a dependent.

Related Information:

• What does "financially support another person" mean?
• What is a "qualifying person" for Head of Household?
• Does a dependent for 2020 have to live with me?
• Do I qualify for Head of Household in 2020?
• Can I claim myself or my spouse as a dependent?