Under what circumstances can each be considered a pest:grouse, hawk, shrew, squirrel, field mouse, cougar, deer, insects, plants?
To add to pohnpei387's post, I'd like to comment on the final item: plants. To me it sounds unusual to hear someone call a plant a "pest." A "weed," certainly, but not a "pest." The idea is the same, though: a plant is a weed when it's growing where we don't want it to grow. Some plants seem to be weeds from the start, such as dandelions, which are beautiful but not valued by most people.
A pest (or weed) is a nuisance, not necessarily harmful but certainly not wanted. So here are some possible answers:
Squirrels are pests when they chew their way into your attic and nest there.
A hawk is a bird I don't usually see as a pest, but maybe someone would, if the hawk were eating up the local songbirds or house pets.
A shrew is a pest when it's burrowing through your garden.
A field mouse is a pest when it finds its way out of the field and into your house, tool shed, garage, etc.
And, for the largest category of pests, insects are pretty much pests whenever they're near us. Most people seem to really dislike insects, especially small flying ones. Some can pose some dangers, such as contamination of food, but mostly we call them pests because they are all over the place and we don't like them.
Any organism can be considered a pest by humans when there are too many of that organism (in humans' opinion) or if the humans feel the organisms are in the wrong place. For example, if deer start to eat our gardens, they bcome pests. If a cougar attacks our pet dogs, they are a pest.
Removing any "pest" from a food chain can cause real problems. For example, if you remove cougars from a food chain, for example, the deer population could increase rapidly and the deer would start eating more from farmers' fields and people's gardens.