Can you have a defined pension and 401k offered by your employer. The 401k has no match.

Yes - the two are not related. One of my former employers had both a defined benefit pension plan and a 401(k) plan, which included a matching contribution. Those were the days!

Sure, I've had several jobs like that, including my current one. While defined benefit plans are becoming less popular, there is no legal reason a company can't have both. But theere is a financial one. It costs much less to administer a defined contribution plan, than a defined benefit one.