Ex-Pet Rat Owner here
Fancy Rats as they are called make great pets that can be taught tricks
Short life span though so they are some what heartbreaking to keep
But Wild Rats can be a nuisance.... Also depends on which type you are dealing with.
Yes and No, to the Rats are disease carriers. The Rats themselves (wild ones at least) have an incredible immune system and it is the parasites that hitch a ride on them that carry most of the nasty things. The rats usually do not catch the disease because of the natural defences.
Rats themselves prefer clean environment and try their best to replace old bedding as it were - yes even wild ones. So they will seek out potential clean bedding where ever the can get it. As well as good food sources, they can also chew through almost anything bar metal (and even then they will have a go).
Sewer rats are some what of an exception, but even then.
Once a female rat has had her litter of potentially 10+ babies the mum is fertile (receptive to impregnation) again with 24 hours. And young rats become fertile within about 4-6 weeks (if I remember correctly). In other words, if you don't sort out the problem you can have a severe population problem within a few months.
Best pest control for rats (in my opinion) a few big mean Cats or Terrier type dogs.
Or decorative Carnivores as we like to call them. If you do have a decorative Carnivore to do the job for you remember to "De-Worm" it regularly!
Traps? Well depends on the trap, but these rats can be devious and unless the trap kills them outright (please note if it can kill a rat it can kill other small animals too) they are likely to find a way out unless the trap is checked frequently.
Just some pointers.
I like rats, but there are rats out there that can be .... Troublesome shall we say