There are a lot of good cases that can be had for your budget.
The best ways to combat dust is keep the PC off of the floor, keep the room clean with regular dusting and vaccuming. This will do way more good than anything else!
I personally do not like filters because they hinder your cooling quite a bit. You can always add more fans to make up for the decreased airflow, but that just makes more noise.
Cases are moslty personal preference, the Lexa Black line looks decent, though I am not too sure about the PSU's they come with. I also see a lot of comments on Newegg about it arriving damaged, cheaply assembled, LCD's that quit working after a short time, etc. May not be the best choice IMHO.
I prefer the toolless designs myself that use latches instead of screws to mount your drives. My Raidmax has these, and also uses locking slider mechanisms to latch your PCI and PCIe cards into place instead of screws.
Pretty handy if you ask me.
A lot of people like the Antec 900.
The case and fans will only stay as clean as the air they move.
However, moving the same air a larger case will stay relatively cleaner than a smaller one and a filtered case will stay cleaner than a non-filtered one. Yes, filters require cleaning and will impede air flow, but your case will require cleaning without filters anyway and what's easier to do: clean fan filters designed to be easy to clean or clean out the fans and the PC innards which can be an annoying undertaking.
As charming and satisfying it may be pulling a sooty, smelly and stringy 6 inch dust-python out of fan innards rinsing a filter is less fuss.