Yes. The reciprocal agreement does not apply to local city tax. It is your employer's responsibility to withhold that tax and remit it to the Ohio city (and that's why they coming after him)**. Technically, if he fails to do so, you should file a city return and pay the tax.
Yes, they should also withhold for your IN county. Some
(but not all) IN counties (but not the state) allow a credit, or partial credit for tax paid
to an OH city. That will come up in the IN state interview.
**Some cities have a construction worker exception and/or a 20 day minimum requirement; but that's between your employer and the city.