Yes. A child dependent has to file a return if their unearned income exceeds $1,050. Capital gain is unearned income. He may be subject to the "kiddie tax", but all of the calculations will be handled in TurboTax.
It's possible you may be able to choose to add the income to your return for your child. However the tax rate may be higher on your return. If you claim him as a dependent he must select that he is being claimed by someone else.
Depending on the child's total unearned income amount, and on your tax bracket, you may be able to elect to report the child's unearned income on Form 8814 inside of your return. See here for requirements and detail: Tax on Child's Unearned Income - IRS Topic 553
What he used the proceeds for from selling the stock has absolutely no effect on it's taxability or reportability. However, as a student he "MAY" qualify for some education credits, regardless of where the money came from. Do be aware however that in order for one to get education credits, they must have paid qualified eduction expenses with *EARNED* income that was earned in the same tax year they are paying the credit. Stock sale proceeds are investment income, not earned income. So I don't know if that will count for anything or not. I suspect not.
Yes and no.
He has to report the capital gain, if it was more than $10,300* (a single person's filing threshold). If the gain was less than $47,750, and is a long term gain, he will not pay any tax as long term capital gains (LTCG) are taxed at 0% for people in the lower tax brackets.
* The numbers are different for people who can be claimed as a dependent by someone else. The filing threshold is $1050 and tax will be due on a LTGC greater than $38,500.
Those numbers assume you have no other reportable income.