Barracks at 200-population functionality

Barracks serve two purposes in the game namely to make children and to increase happiness.

At 200 dwellers:

• can a dweller still become pregnant?
• Would there be any increase in happiness?

Since the other answer does not answer the giving-birth aspect of dwellers becoming pregnant:

Vault dwellers can become pregnant when your vault is it's maximum capacity, but they will be unable to give birth. When tapping on the icon to make your vault dweller give birth she will say "There is not enough room in the vault for the baby" or a similar phrase.

This is true whether you've reached your individual vault's barracks capacity or the overall 200 vault dweller maximum population. If you lower your vault's population by killing an existing vault dweller during an attack, fire, or wasteland exploration your vault will have enough room to allow your female dweller give birth.

Regardless of whether a dweller is able to give birth or not, becoming pregnant will automatically provide the normal boost to 100% happiness to both the mother and father.

Dwellers can still become pregnant, which can be a problem because pregnant dwellers will not fight invasions, and can not be sent out into the wasteland. They will remain pregnant until room is cleared for them (by removing dwellers). The happiness increases will remain in place, because there is always a reason for dwellers to get depressed, from them not doing the job they're best at (being in the wrong room) to having a dead body nearby.