Does even the odor of cigarettes on clothes cause problems for a child with asthma?

Yes. The odor itself isn't an issue but where there is odor there are chemicals from the smoke that can cause all sorts of problems for people.
Yes. It definitely can. This second-hand smoke you are referring to can often be a major airway irritant and trigger for coughing and wheezing in a child. There does not have to be direct exposure to the cigarette smoke.
Yes. Cigarette smoke is essentially a poison or toxin that those with asthma are especially sensitive to. Children are more affected than adults. As an analogy, imagine trying to breathe through a straw (pediatric airway) with inflammation and mucus in it versus a garden hose (adult airway) with the same. There are many options to help you quit smoking, so be sure to discuss these with your doctor.
Yes. Yes, that's called "third hand exposure" and is quite injurious to children including asthmatics.
Yes. Being near or holding a child while wearing clothing that has been exposed to cigarette smoke can increase breathing problems for any child - especially with one already diagnosed with asthma. I recommend quitting smoking for optimum improvement for the child's health. Smoking outside, wearing a heavy jacket, and changing clothes can all help but are still putting the child's health at risk.
Yes. Absolutely. It is not ok to smoke outside and think that you are protecting your child. The smoke and other chemicals can still be transmitted to your child. You have to quit to help prevent asthma symptoms.