Age is not the issue. All who wear diapers are to keep themselves from the tub, and all who don't are allowed in.Toddlers are welcomed to opposite gender bathrooms accompanied by the proper gender parent or other adults.
At large facilities, there are often several tubs with different tempertures. On the other hand, in smaller facilities, they sometimes fail to keep the right temperture and the tub is often too cold. The "ordinary temperture" of an onsen tub water is supposed to be 37-42 degrees centigrade.
You can request private bath for any reason you can think of, where you will be filling the tub yourself, but when you want to have a ready-filled public bath temperture arranged, you need to talk to the manager. If there is no one else using the bath, the manager would usually be happy to adjust the temperture for you. However, keep in mind that this may take time. So it would be best if you take a peek in the bathroom before hand to see if the conditions are ideal. Of course, if there is anyone bathing, be sure you enter naked, too, as entering with clothes on may embarrass others who are naked.
Have fun!
