Is the Home Owner liable for an injury if the licensed contractor is injured and is exempt from worker’s compensation?

As owner of the contracting company you probably don't qualify for w comp coverage. I'm sure your contract makes you as contractor responsible for complying w all applicable laws, including WC for your employees. There is also probably an indemnity clause requiring you to indemnify homeowner from WC claims by your employees. If you are injured by your own fault, w/ no negligence by home owner, the home owner cannot possibly be liable for your injuries. WC won't apply because you are an independent contractor, not an employee of home owner. If you are WC exempt, that's your own voluntary choice, making you non covered by WC even as an employee of your own company.

This is usually true for unlicensed casual employees only. The law on the subject is very complex and depends on factors not answered in your question. I would urge that you seek consultation with an experienced personal injury attorney.

No, if we are talking negligence. However, some home owners policies have coverage for casual worker's comp benefits. Ask to review the policy.

Since you are an independent contractor there is no workers' compensation coverage involved. If there is no negligence by the homeowner then there is no liability.

No they are not liable unless there is negligence. However, under most home policies there is a no fault med pay policy for injuries occurring in the premises up to usually about $5,000.00.