Noobiest Noob

The downside of a soldier class is this; they will always lose a firefight with a heavy, run out of darts before a scout and have less backup than a medic (as people don't constantly run up to them for respawns that the medic enjoys). While a soldier class is definitely fun and balanced I have a feeling that its just too bread and butter for most people. After all, the idea of Classy Wars is to have a class, and the soldier negates that. Of course it offers greater variability in the team setup, so presenting it as an option can't hurt at all.

If you need a 'special ability' for the proposed Soldier class, why not make them the only ones able to take and/or hold objectives (or flags, or what have you)? I'm more or less stealing this from 40k, or at least the older editions that I played.
I'm not too concerned with Soldiers 'always' losing to a Heavy, as that's seems to be the point of that class. I think that two or three taking on one Heavy is a roughly fair fight, depending on cover available. The same with running out of darts before a Scout.
You bring up a good point with the lack of support, though. Perhaps it takes a few losses for players to learn that giving each other support and back-up is a good idea? In any case, I agree with you that these issues are mostly important for smaller games. I'm sure it's possible to break the system with a larger group, but since everything is still being developed, it's going to help playtesting.

To be fully honest, I was working on something similar to this, but much more complicated (and not in a good way). I'm just glad I'm not the only person that thinks something like this is worth exploring.

I think the 'Classy War' is the Nerf version of a 'boffer LARP', and would probably work best when played in a similar fashion. Having a regular group, as opposed to just a random gathering, with an objective based structure, instead of a re-skinned deathmatch.

With the disclaimer that I have almost no experience with Nerf wars in any setting other than an impromptu game at birthday parties, I would suggest editing the classes slightly. Using the Scout as a baseline, I think limiting them to two weapons is fair. I would say nothing needing two hands, but if your players are smart, the should figure that out quickly. The ammo limits may scare off some players from trying the role, but the ability to collect ammo and restock teammates is powerful. To highlight that, I think adding some limits on carried ammo would be required. The unlimited capacity of the Scout has been mentioned, but what about the other classes?
The only issues I see with the Heavy are the potential for cheating by not calling or counting hits (and if you have players like that, find new players), and the ability for a squad of them to completely dominate the game, especially with high capacity, long range or high RoF blasters.
As for the Medic, I think limiting them to springers only is a little underwhelming. The focus of the class seems to be on tactical positioning rather than scoring kills (or whatever), so instead, I think limiting the class to pistols would be better. They can still mount a defense if attacked, but won't feel the need to rush out to go for the kill, which I believe fits the flavor of the class.

I'm not going to bother trying to list possible objectives. Stealing from a miniature war game rule book would probably be a decent start. Of course, if a group is forced to do more standard game types with the class-based structure, I would put heavy limits on how many of each class can be fielded, and perhaps add a fourth, generic Soldier class, with one hit point, probably using a primary and sidearm, and having the ability to run and nothing else. The limits would probably vary based on number of players and personal preference, but I would suggest no more than two Medics, and only if playing a large game. Any number of Scouts would probably be ok, and Heavies ...I think 25% or less of the team size should be allowed. The generic class would fill up the rest of the team.

Again, take my insights with a grain of salt. I'm commenting more as a member of the tabletop community rather than the Nerf community.