Should I have my child checked for asthma if he gets lots of colds and coughs?

Yes! It really depends on the frequency of the respiratory symptoms, but if he coughs (on average) more than two days a week or more than two nights per month, an asthma evaluation is warranted.
Yes. If your child gets lots of colds that "go to his/her chest" or the cough tends to last a long time whenever he/she gets sick, then that is more likely to be asthma. Definitely get them checked, as if so it should be treated to control the ongoing inflammation in their lungs.
Maybe. Colds and coughs are common and most are not asthma related. The way to know is have your pediatrician check your child during these episodes of cold and cough. I'm sure your doctor will be able to tell.
No. Asthma means that he wheezes audibly beause of narrowing of the broncdhial tubes. Lots of colds and coughs in my experience almost certainly related to "junk" diet. Please see my blog "oxygen the spark of life" that you can google.