they both have fur?
Not recently. Cervidae is the deer family. Rodent is the Rat family. When the dinosaurs died only Mustelidae's survived. Over the millions of years everything was in a arms race against each other.
The most common rat is the brown German rat.
deer droplings
Some breeds of dogs are named for things they were used in hunting for. A deer dog, or deer hound, would have gone on hunting trips for deer. There are wolf hounds and rat terriers, etc.
Yes, the noun 'rat' is a common noun, a word for any rat of any kind, anywhere.
Unlikely, unless the deer is wounded. Rat poison is an anticoagulant, and can only kill if the animal begins bleeding due to cuts or other injuries. Rat poison works on rats because they fight frequently and once injured the poison causes them to uncontrollably bleed to death.
Sounds like a deer mouse.
Yes, the noun 'rat' is a common noun, a word for any rat of any kind, anywhere.
Capybara
Yes, the white tailed deer is common there.
An Asian black rat is an alternative term for the black rat.
They're a common rat also known as a sewer rat.