Just the opinions of someone that doesn't normally play this type of game
Maybe consider letting the passive buffs(extra gold/hp regen/increased xp/increased mobility/increased reach/quicker reloads/ extra hp/ etc.) be what you buy from the store instead of the other weapon types, don't lock those or have them be unlockable via quests, but make the purchases of each buff permanent each level getting more expensive as you purchase them so that you have a sense of progression.
The sweater is the only good starter weapon in my opinion if you can get it to level 2 early on the timer/rng doesn't seem completely unfair. The fact that items(xp drops) despawn if you leave the area too much makes pulling enemies away too impractical and this is basically the only weapon that lets you remain relatively safe when doing so.
The sword and bow feel like they should have their attack patterns swapped, the bow always shooting where you're headed encourages charging the opponent(close range approaches), whilst the inability to change the attack direction of the sword swings and the honestly decent range they have encourages you to kite enemies and keep them at range(long range approaches).
The wand and boomerang are great support weapons to have as a bonus to your main weapon right away but don't really make for good main weapons until they've been upgraded a decent amount, the timer makes taking your time early on a terrible idea.
The fire weapon is just terrible, maybe give it's attacks range more on par with what the sweater has and leave the rest of this weapon as is, the slow random targeting doesn't work with the giant swarms of enemies and the items despawning if their too close to the edge of the screen at all, and would work to keep this item from being overpowered even with that buff. As it currently is you're actively better off never purchasing this weapon so that you can get literally anything else to come up as an option when the upgrade selection screen comes up.