No and no.
Most important is that you understand that each HSA is owned by one person only. There is no such thing as a "family" HSA. There is a family limit to tax-deductible contributions, if you are covered by family HDHP.
You and your spouse may each have a separate HSA to which you can make $3500 of contributions for 2019 ($3550 for 2020). If covered by single HDHP as well, your son can open his own HSA account and contribute up to the same limits.
