Yeah, to consistently get any iridium you probably want to reach level 50+ at least.
I wouldn't necessarily work on the combat. Getting from 5 to 10 would probably take a while, and IMO nothing in levels 6-9 is really that important unless you want to try using explosive ammo (iridium band at combat 9 is very nice but you'd have trouble finding the iridium to craft it without going into SC for iridium anyways). A few extra HP probably won't make too much difference against SC monsters. But if you do want to work on it, try going into mine levels 85/95/105/115 repeatedly and just killing any void spirits near the entrance, that's probably the fastest way to get combat XP in the regular mines.
If you can buy the lava katana, that would be great. It's definitely possible to do Skull Caverns with the steel falchion, but it's quite hard.
If you have a hammer, you can do several strong hits at once by right-clicking and then mashing left click. The best hammer you have access right now is probably the lead rod or maybe the wooden mallet, both of which will not do enough damage to kill enemies in any reasonable amount of time, but they'll knock back flying enemies quite a bit and then you can run away (or switch to your falchion if you really want to fight them, but with your weapon setup you should mostly be trying to dodge enemies).
The other option is to keep killing enemies in the regular mines, I guess - after you reach level 120 you have a 0.5% chance of getting a prismatic shard when you kill any enemy, and then you can get the galaxy sword with that (and the galaxy hammer if you can afford it).
Speaking of dodging enemies, you really want as much speed as you can. Once you get to the desert, you can trade rubies and diamonds to the trader there, to get spicy eels and triple shot espresso (note that espresso, being a drink, stacks with food). You'll need about 3 spicy eels and 5 espresso for a full day. This will give you +2 speed, enough to outrun the flying serpents, which should hopefully help you avoid the weaponry issue.
Your rings are not terrible IMO. Another glowstone or magnet ring in place of the glow ring might be nice if you're using bombs, so you can collect all the stuff dropped by the bomb faster. 2 glowstone is basically the same as 2 iridium except without the +10% attack, and we've established that you shouldn't be trying to kill things anyways.
As Ereo said, bring lots of food, and don't let your HP get below half. Any single hit from most enemies can take away around a third of your HP, so keep an eye on it! If you've got some money, buying salads from Gus is very money-efficient. Salmonberries or blackberries are also good money-wise, but you have to eat a lot to regain any appreciable amount of HP, which is annoying.
For bombs I'd say at the very least you want 20-30 regular bombs, preferably 50+. Ideally you barely use your pickaxe (this is the case even if you somehow have an iridium pickaxe). You can of course go down with less, but you definitely won't get as far. Assuming you have mining level 6, maybe spend a day or two in the iron levels of the mines? Going up and down repeatedly from level 40 should net you a decent amount of ore, and killing dust sprites should get you some coal (and maybe that prismatic shard).
But based on Miretar's comment "Previous visits had me out of those caverns in about 15 seconds with my tail between my legs" I suspect that the 40k to finish the Vault bundles would be better spent around the farm, rather than trying to access the Desert anytime soon.
Staying in town and spending the money I would have spent on the Vault so that I can irrigate my most precious crops with sewer sprinklers is clearly the correct plan! (Also, this game is tremendously amusing at times...).
