is it a normal thing if dogs cough and vomit?

I don't think it's all because of your cat's hair. Chihuahuas are very sensitive to variances in temperature and weather condition. Since they have very narrow & thin windpipes, you may find them sometimes panting like people suffering from asthma, a kind of human respiratory diseases. Pay special attention in taking care of them during days in low humidity (dry weather). If neccessary, breathing in vapour will help to release the discomforts of windpipe contraction.
Visit the vet immediately if situation is getting worse.

Coughing and Sneezing is the symptoms of 狗瘟.

If your dog 行得食得走得．ｄ便便仲好正常,
Then you dog is 輕度狗瘟.

You should take your Chihuahua to see VET
because has 狗瘟ｄ dog have only 50% chance of healing.
But VET is very expensive. Think about it.
If it is 狗瘟, then you will need one year to heal it.

I am afraid that cough is not a normal symptom for the dog. You'd better take he/she to the vet and have a check about it.

Sometimes, the dog vomit because they ate many rubbish/dirty objects (such as hair) which they can't digest.

Hope your dog will get well soon.

It's hard to say for such sympton. For a better result, please take your dog to a vet for further checking.

I think so, it happen to my dog too, he ate a plastic bag coner