There is no specific name for a cage for rats, its just based on what the housing is. So a cage for rats is just a cage. (Just like a hutch for rabbits is still a hutch even with no rabbits in)
If they are pet rats kept in a clean cage, then you can't catch anything. Otherwise they may have fleas.
Terrarium
Regardless of gender, a rat is only as clean as it's habitat. If the cage is kept clean there will be no issues of smell.
No. Rats should be in pairs or more. They become lonely and depressed if kept in a cage all by themselves.
They weren’t kept in wooden cages, but in chains.
yes, dwarf hamsters can live in cages with tunnels in fact it is best to have one with tunnels so they can crawl around :)
Not all rats are afraid of cages. My rats have always loved their cage, and when let out to play would wander in and out at their leisure. If you have rats who are afraid of their cage maybe there is something in the cage they do not like, maybe the cage is too small, maybe the cage is not comfortable enough or in a good position. Maybe there is something near the cage that scares the rats or maybe they do not get enough playtime and then resist going back into their cage. There are many reasons why a rat may be fearful of their cage, so there is no definitive answer.
Snake, deer but a deer needs to be kept in a big open cage probably in a field, lots of birds are diurnal, a horse but they need to be in a cage like a deer.
No.
no
yup, wild ones almost always do. And pet rats have their cage.
A battery cage is a small cage in which a hen is kept to lay eggs.