The Apple iOS and OS X Japanese keyboards suggest emoji for words along side the Kanji (your ねこ example) and that seems to be the main way the people around me enter them (I'm in Japan). It's easier to use since you don't have to go paging through 20 pages of emoji hunting for a particular one. I have yet to use Android since they added emoji support so I'm not sure what's going on there.
Are the people around you using iphones? The iOS Japanese keyboard suggests a lot more emoji during normal input than the old pre-smart or Android Japanese phones I've used....
Anyway, the main point I'm trying to make is that the method of Japanese language input doesn't seem to adequately explain why emoji are so popular in Japan, because the support for conversion-based emoji input is haphazard even in Japan.
