As others have already replied, there are lot of RPGs out there, and if your players are willing to try out new systems, not all games need to center on the use of violence as the primary mode of play.
Most RPGs include violence because it's a quick way to raise the dramatic stakes in the story. Risking life and limb is exciting. Indiana Jones violently battling Nazi's for control of the Lost Ark is a lot more dramatic than deciding who gets the Ark over a game of gin rummy.
RPGs can be like roller coasters, in that they offer us the thrill of danger, without putting us in actual peril, since it's just a game, and the actions are only played in our imagination. Violence is an easy way to provide that thrill.
The employment of violence for dramatic effect can become rote and cliche if overused. For example, most TV action shows of the 50s, 60s and 70s would include a ubiquitous fight scene, thrown in as padding to "spice up the action", even if it didn't really add anything to the story. On the 60s Star Trek, this usually resulted in yet another torn shirt for Captain Kirk.
With D&D, maybe you could propose changing the application of violent scenes in favor of more roleplaying. This would allow you to set this expectation with your players.
D&D is at it's heart, and action-adventure genre game, so it's got a lot of mechanics that are action-oriented over RPing. You can set story awards for resolving problems in non-violent ways. You can also make encounters that can involve combat as just one of many options. For example, you can cross the Troll Bridge by fighting the Troll, outwitting the troll, bribing the troll, solving the troll's special problem (maybe he wants to marry another troll, and the PCs arrange this), etc. Maybe the non-violent solutions are better than the combat option. As long as it's a challenge to solve, it's worth the same XP.
Some examples of RPGs with mechanics supporting non-combat solutions include: Apocalypse World, Robin Laws' "Hillfolk", Fate, and Burning Wheel/Mouseguard (these actually have a social combat system, with social maneuvers, and rounds of verbal "combat").
You can also include action scenes that don't involve combat. If the players are traveling on a boat, they could encounter a storm, and depending on how they work together to weather it, they could suffer escalating setbacks (including having the boat sunk). Maybe they have to pass through an obstacle like the Scylla and Charybdis -- an action scene that combat won't solve. You could have a chase sequence where they don't want to injure the person, but encounter dangerous obstacles during the chase.
As far as the younger player having a meltdown, for session zero, I'd set the expectation that PC death is a thing, it's part of "winning" and "losing" a D&D game, and that the group's social contract includes the expectation that players handle "losing" a PC with the same maturity and good sportsmanship as they would losing a game of monopoly or checkers. You can't win em' all, but you can always try again. Flipping the board when you lose too many times will not get you invited back to play, and the same standard of behavior applies to RPGs.