Can an antiviral and antibiotic together be taken together?

Usually safe. Most routine antibiotics can be combined with antiviral medicine when used for short periods of time. Taking any antibiotic for >2 weeks usually requires close supervision by a doctor to assess for side effects, toxicity & medication interactions. Special antivirals (ie. Hiv medication) require closer monitoring. People with kidney, liver or bone marrow problems are at higher risk for toxicity.
Yes. For patients that have a viral infection with secondary bacterial infection, this is often done.