Could radioactive seed treatment cause problems for my pet?

No. When having seed implants for the prostate the rules are very minimal. Usually consisting of do not allow children or pregnant women on your lap or very close to you except for brief periods. Depending on the seed type this is for 2 to 6 months. Calculate the age equivelant of your pet and follow the same rules. For example no puppy or kitten on your lap for the same time period.
Not likely. If you follow the same rules about distance and time to be apart as for people, there should not be a problem.
Yes. Agree with dr. Daneshfar's reasoning but as he explains apply the same you would apply to humans to your pet.