Walaikum Salam. Yes. You will need to pay zakat. Income tax is paid to the government. Zakat is to be paid to muslims. See IslamQA for detailed list of categories.
Answer to your next question is, you need to either make up for the missed days if you have ability to fast at later stage or pay your Fidyah (Expiation) if it is terminal illness, or inability to fast because of old age.
Another common misconception amongst muslims is that zakat is to be paid in Ramadan. Your Zakat year should begin on the date your wealth equals or surpasses the Nisab, that will initiate a new year for Zakat. You will then see a year after this new date whether your wealth equals to or surpasses the Nisab. This is referred to as your ‘Zakat due date’. Zakat is due immediately upon your 'Zakat due date'. If you’re not sure when this is or you’re paying for the first time, simply work out the day you first owned wealth over and above the Nisab threshold (if you don’t know the exact date you can estimate). Muslims pay in the month of Ramadan because the reward is multiplied.