No. Sage is a background.Scholar is a background.

Most fighters, rogues, clerics, and wizards are adventurers either.Most scholars are not adventurers. If they were, they would need to have some sort of other skills, be it fighting or magic. Adventuring scholar is a character of any class with appropriate skills.
Yes, a scholar would be trained to fight. Just not at level as a samurai or evoker.
Holmes not a wizard or rogue. Leonardo would not be a artificer.Leonardo was not an adventurer and a fantasy Leonardo would have absolutely been an artificer. The guy was one of the greatest geniuses who have ever lived and dabbled in everything. If he had lived in a world where magic was learnable by studying, he would have absolutely done so. And same with Holmes, fantasy Holmes might be a divination wizard. Though if you wanted to do a non-magical version, there literally is a rogue subclass called 'inquisitive' for that.
Again, the things you want are not classes, and are already doable in the game. And were they classes it would limit what the other classes can be, Now you can be a scholarly fighter or a rogue, but were scholar a separate class this wouldn't really be the case any more.
Many complain about grid filling in 4e but D&D fans often force concepts into existing classes. Rather than trying to contort concepts into existing classes, these lost skilled concepts shoulld have their own class.