As someone with a father sort of like this, (though he at least never chewed me out for it, but made an effort to watch with me if I had something I really wanted to show him) the ones I remember him actually sort of liking were Cowboy Bebop and Tokyo Godfathers. Millennium Actress is quite lovely, but some might find it a bit confusing to follow. Summer Wars was great fun too, with a nice family oriented kind of story and fun adventure.
Getting more into the realm of a bit more violent, Jin-Roh is a bit bloody, but is drawn really realistically and doesn't have the standard anime fanservicy pandering (if I recall right), so that could be a good one. I adore Monster as well; great suspense thriller to me, but it's also a 74 episode commitment.
That's what I got off the top of my head.