Yes, it's contagious. Chlamydia germs can also live on the throat, not just in the penis and in the vagina/cervix. The idea is to avoid oral sex with people carrying infections, until the infections are cured by medications. If a person with a sore throat goes to see the doctor, he should tell the doctor if he has been doing oral sex because the antibiotic for strep throat is different from the one to treat chlamydia.
Usually not, but... Chlamydia doesn't take well to the throat -- even with oral sex, it's a lot less frequent than gonorrhea of the throat, for example. However, you shouldn't take the chance. If the other person knowns s/he has chlamydia and not yet treated, s/he should not be having sex of any kind, with anyone; and you should avoid having sex with such a partner. A little common sense goes a long way!