No, you shouldn't, it's very rarely pronounced as it sounds too forced. Good luck with Czech!
I can remember my Czech teacher pronouncing it when she was speaking very slowly and trying to be very clear, so I don't think it is silent in the way that the k in the English word knife is silent. It's certainly not pronounced in normal speech, though.
According to this message board http://www.myczechrepublic.com/boards/viewtopic.php?t=7779&view=previous the 'j' is omitted in colloquial speech, but is usually pronounced in standard or formal speech.
Formally it is correct to pronounce with "j" and if you ask some Czech to say this single word, he would probably pronounce it with "j". But in sentences in common speach it is often omitted since it is faster and easier:) But you can pronounce it in both ways, you will be understood and you won't sound weird.
Why is this "jsem Kateřina", and not "já jsem Kateřina"? Are both correct?
Yes. Generally, the pronoun can be dropped, if it is clear from the verb form. If you specify the já, then you are emphasizing it, e.g. You are only a student; I am a teacher.