(Credit for these tips goes to MaxisSparks @Reddit)
Industry delivers freight.
Commercial and Trade Depots receive it.
Working industry sends out trucks that deliver freight. If the freight successfully delivers, the industry gets 1 happiness. Freight is normally delivered to commercial buildings. They have a certain amount they can take per hour, and consume what they have slowly over time. Trade Lots with the freight module can act as a freight delivery site in the same way.
There is no connection between the Depot and Commercial. They both just consume freight.
Commercial buildings don't need freight to sell goods. Goods sold regenerate independently. You can build an RC city, for example, and you'll have to deal with significant unemployed, rather then unhappy shops.
Freight doesn't make commercial happier, just industrial. Selling Goods makes commercial happier. Goods aren't related to freight.
Each building can only hold a limited amount of freight
The depot empties slowly over time. If you want to store more freight, build more freight modules.