1) The food thing: Check the SPECIAL stats of your dwellers, and assign those with a high Agility to the kitchen. When you grab a Dweller and hover them over the kitchen you will either see (A 3) if there's a free spot available for them to work, or (A+2) if their Agility is higher than that of another Dweller working there. They will then swap places.
If your Dwellers all do not have high Agility scores, try equipping them with armor that raises it. These can be found by sending other Dwellers out to explore the wasteland (not the same as going on a quest! see below). Also, you can build a gym in which Dwellers can train their Agility. You will need at least 24 Dwellers to unlock that building.
2) When you select a quest, you'll see what level your Dwellers need before you can accept it. I assume your 'missing adventurer' is not of a high enough level.
Note that this is not the same as exploring the Wasteland; You can grab a dweller and put them on the ground outside the Vault to have them go exploring. Equip them with a weapon, armor and Stimpaks/Radaway when you do though. The Wasteland 's a tough place...