The coughing fit and loss of appetite require he see the vet.

Throwing up soon after eating "could" be caused by eating too fast. My RB Chestnut would inhale food like a vacuum, when he threw up it came up whole, not chewed at all. I had to feed him small amounts at a time, wait 10 minutes, give him more. Then he did OK. Chestnut was a foundling, and had to struggle for food for a time when he was a baby, so he had not learned to temper his intake; it was "get it while it is there before it disappears!"

But be sure to mention this issue to the vet as well, in case this is related to the coughing and loss of appetite. I have one cat with a minor heart murmur since a kitten, 6 years later it is still not a big deal. The coughing can be part of a 'kitty cold,' an upper respiratory infection, and nothing to do with the murmur, so the vet needs to see the kitten to sort this out.