If your friend is still within the time period provided by Ohio law he can appeal. The points about the trial that you mention would not necessarily result in a successful appeal, but only a lawyer familiar with the entire record could evaluate the case properly.
Deadlines for initiating an appeal are typically very short (the thirty days from judgment iallowed by llinois is typical but some states allow a little more time and some states and the federal courts allow even less time) and there may be important preliminary requirements, like some sort of post-trial motion. This varies a lot with state law. It is the duty of the trial attorney to file any post-trial motions and to be sure that a timely notice of appeal is filed, but your friend must tell his attorney that he wants to appeal.
An appeal is definitely not a do-it-yourself project and he will need an attorney. Whether to ask the same attorney who tried the case to handle the appeal or whether to retain new counsel, perhaps an appellate specialist, is your friend's decision. If he cannot afford counsel on appeal the court will appoint an attorney to represent him.
Under Ohio law, a defendant has 30 days to file a notice of appeal after the filing of the sentencing entry. Missing the 30 day deadline can forever end a defendant's rights to appeal -- it is possible to file an appeal beyond the 30 days, but greatly increases the chances that the appeal will be thrown out of court.
I recommend contacting an experienced attorney who handles appeal as soon as possible. A third degree felony is punishable by up to 5 years incarceration and domestic violence and child endangerment convictions can damage a person's career for the rest of their lives.