"These dogs are searching for the food" was wrong... Why?
Is "nämä" just the plural of "tämä"?
"tämä" - this
"nämä" - these
Yes, that's correct.
These dogs are looking for the food ??
Yup, that is also correct, since we have no context to know whether it's particular food or any food. Did you report it using the flag?
Groke: to gaze expectantly at one with food in order to entice them to share their spoils.
E.g. Whenever I sit at the table with pulla, my sister grokes at me.