Yes. I bought "My Pillows", which are machine washable and dryable in their entirety.

My cat jumps up on the bed right across the top of my pillow, and usually barely misses my face. I have even found grains of kitty litter on top of the sheets, and I want my pillow to be disinfected and washed at least once a month, and the sheets once a week.

Yes. All bedding needs a wash.

actually, no, but periodically put them into the drier at a high temperature to kill any organisms.

I wash the pillow covers, but the pillows themselves I just throw away and replace if they get too gross.

Yes, every week.