Bodily wastes are a taboo subject in most cultures, but not that much in Japan. In fact, the subject of poo, or the lack thereof (as in constipation) is discussed fairly openly in polite society.
There are many different ways of calling poo. When it's on the hard side, it's called うんこ (unko), while the softer kind is called うんち (unchi). These are both informal and rather cute terms; the formal term (the equivalent of feces in English) is 便 （べん ben) or 大便 (だいべん daiben). This combines the kanji for 'big' with 'ben'. Another word for it (a bit rude) is 糞 (くそ kuso), also read as ふん fun. くそ is also a swear word, like 'shit' in English, but milder in terms of shock value.
On J-List they have several cute unchi (poo) goods, including a pink unchi pencil topper mascot. Kawaii poo: Only in Japan?
Here's a mysterious yet colorful video about 'The unchi of love'.
Constipation, a problem that is considered to be quite critical for ones health, is called 便秘 (べんぴ ben-pi). This combines the kanji 'ben' with the one that means secret or hidden. So constipation is your body secreting or hiding your poo. Makes sense, doesn't it?
The opposite problem, diarrhea, is called 下痢 （げり geri). Some people get snickery about the male English name Gary, since it sort of sounds like 下痢.
The formal names for urine are 尿 (にょう nyou) or 小便 (しょうべん shouben - 'small' ben as opposed to 'big' ben for poo). Informally it's called しょんべん (shonben) or (especially by kids) おしっこ (oshikko).
As far as I know though, there are no kawaii pee mascots.
(Edit: Apparently there is a poo and pee doll pair, though they are Scandinavian, not Japanese: see a picture (thanks Sile...I think ^_^;)