All the weapons come launch are supposed to be side-grades of the starting weapons. Some will be superior in certain situations, but as a general rule you shouldn't be able to buy an overall advantage through the real money/Auraxium store.
Currently this isn't quite balanced, as some weapons are buggy and others are pretty much required purchases (MAX AA for one...). Hopefully this will be resolved before launch. But certainly, depending on playstyle, you can be entirely effective without any unlocks. You will just be less (and in a couple cases, completely in-)effective in a few roles till you can unlock a suitable weapon for that.
Auraxium seems to flow reasonably well, but more-so the more you play. The two main sources are; 1) controlling Auraxium generating bases and, 2) fighting in those same bases. Any experience/points gained in any base will return some of that base's resource. Given the Auraxium bases are the largest, multi-point bases, designed for the longest and bloodiest fighting, a solid amount of any player's fighting should be returning Auraxium.
There is some debate right now about how passive gain should work, and it seems like some passive gain should exist even when offline, if only to stop people idling AFK in the warpgate, filling the servers. At least the random EU1 resets have been wiping these people out.
As for play time, I'm not sure how many hours I've played but so far I've bought two AA guns at 8k each and have saved up around 12K since then. The weapons that have been put on sale seem to be between 7.5k and 25k. Given I've played more than average I'd say that's close to 3-4 weeks worth of 'normal' play, without idling for passive gain.
It will also depend somewhat on your factions map control, which could put you at a strong advantage or completely starve a weak side. That said, balance seems to bounce around very quickly in this game, and only changes the passive gain. Being outnumbered should actually improve your active gain, although you would need to keep the fighting in the right region to get the resources you want.
The only straight upgrades in game are supposed to be through certifications which only come through playing. These do give a straight improvement and advantage to players who have been around longer. They come both with experience and straight time, earning a few just for your character existing, even while offline. Again, fighting returns them a lot faster, both through experience gain and a straight cert point for every base captured.
Of course, certifications themselves are sidegrades compared to other certifications, and you are very limited as to which ones can be active at the same time. With the armour certs I listed before, a long time player will be able to choose any one of those for the situation they are about to throw themselves into, but only one for each time they equip their character. This gives them an edge over a player without any armour enhancements, but it's still about situational advantage.
The Word of God is that a fully certed player should have something like a 20% advantage over a new player. I think it's a little more than that right now, just because of stuff like the C4. My upgraded HA can always insta-kill a MAX who isn't paying attention or who I can taunt into chasing me around a corner. A new HA will have a much harder fight.
For people who play/know League of Legends, certs are like runes. Having runes gives you a decent edge over those without, but once everyone has them it's more about choosing the right ones for your character's kit and hoping it lines up with what the enemy do in that particular game. It also looks like some might be tied to level, such as the leadership ones.
The weapons, on the other hand, are like new champions. More expensive to unlock (and the game's main source of real money), but supposedly all balanced against each other and, where it makes sense to compare them, they are only sidegrades of each other. One may dominate in some conditions, but overall there shouldn't be any that rule all. How this works in practice... well, here's hoping SOE get better results than Riot. The current situation, without free starting AA weapons, is like never having a free AP champion in the rotation.