Wendy,
Tatami is not ideal for putting furniture or babies on, but locals compromise, because quite often it's the only room you've got.
Typically, parents would take off paper doors off or turn them inside out when there is a little one at home. They would also lay a sheet (such as quilt) on the tatami floor. Otherwise, paper doors often end up looking like a haunted mansion, and there is no limit to the vomit and urine that the tatami sucks.
However, I used to use our tatami as is when my child was young. I would wipe off any dirt as soon as possible, and spray some etahnol on it and it was okay. But still, with yours and other's babies crawling around and toys rolling back and forth, sooner or later a bunch of rush (the material for tatami) bits started scattering all over. Needless to say, you need to make sure your tatami is vacume cleaned often enough in order to avoid mites.
That said, tatami and paper doors are expendables anyway. You are expected to exchange them once every few years, and if your home is rented, your deposit will go on all the maintenance including the exchanging of the tatami and paper doors, whether you have kept it clean or not.
As for furniture such as desks and beds, locals do put them on tatami, but often place a rug underneath to keep the damage as little as possible. But still you end up with dent due to the pressure caused by the furniture legs.
One important thing to keep in mind. If you lay a futon directly on the tatami in the traditional Japanese way, be sure you put it away when you are awake. If you lay futon at all times, the tatami will become damp, therefore attracting mites and even giving damage to the wooden layer underneath if you leave it there long enough.
I'm not sure what you mean by "specialized furniture for tatami rooms", but basically, in a tatami room you are supposed to sit and sleep on the floor (using zabuton cusions and futon mattresses). So tables and desks need to have shorter legs. Typically, these short-legged furnitures are put away when you lay the futon to go to sleep. But again, it's up to you to use chairs and beds. By the way, IKEA does not really sell Japanese style furniture.
Why do modern 4bdrm apartments have tatami rooms then? Because some people just feel at home with tatami.
Just for reference, here are Googled photos of beds on tatami rooms.
http://images.google.co.jp/images?q=%E5%92%8C%E5%AE%A4%...