Take the mats outside and clean empty them into the garbage, then sweep that way the biggest pieces go straight into the garbage not to the sweeper. As far as dust, you can buy dust free liter, as far as the filter - contact your local contractor, they can tell you what is available and how much it would cost to put it in. Good luck

Check out the air filters listed below. I have the Hunter 30245 in my bird room and find it runs quieter than the lower end models.

Prices vary so do some research. I think I purchased mine at HD (or Lowes?) for less.

Here is a link that might be useful: Air filter

I use a Bissell rechargeable sweeper for quick clean ups of litter and cat hair. The thing is not a deep cleaner, but it picks up litter and surface hair really well. Good for stopping the spread of litter into the rest of the house and for quick cleanups of spilled cereal, etc. in the kitchen. I have had mine for about 4 years and it still charges good and works fine. Easy to empty the dirt holder, too. Amazon has it, as well as Target and HD.

Here is a link that might be useful: Bissell Sweeper