Can I start my own business after completing web designing and development course?

Yes, you can do it, but it requires high level of computer knowledge and a lot of funding, because the software needed are expensive and you also would be needing an expertized team to work with you.

If your job's terms and conditions allow, else that may turn out to be a legal violation. However, you could start it up and get it registered in your family members name but transactions of that business in your name may not be suitable if owing to your job terms. Else you could.