There is software that allows you to draw the characters on a touch screen, but it's still slower than typing and most people either type the words phonetically (e.g. in Pinyin) or use something like Cangjie which is based on the shape of character components. Shape-based methods are harder to learn but can result in faster input.
For Japanese there are keyboard-based romaji and kana based input methods.
The correct Hanzi/Kanji is either inferred from context or, as a back-up, selected by the user.
Almost nobody uses kana input in Japanese. There are specialized systems for newspaper editors, stenographers, etc., that are just as fast as English-language equivalents, but learning them is too difficult for most people to bother.
Huh? The standard Japanese keyboard layout is Hiragana-based, and every touchscreen Japanese keyboard I've used defaults to 9key kana input. Do people really normally use the romanji IME instead?
Yes. Despite the fact that kana are printed on the keys, pretty much no Japanese person who is not very elderly uses kana input. They use the romaji (no N) input instead, and that's the default setting for keyboards on Japanese PCs.
You are right that the ten-key method is more common on cell phones (and older feature phones had kana assigned to number keys), although romaji input is also available.
For Japanese there are keyboard-based romaji and kana based input methods.
The correct Hanzi/Kanji is either inferred from context or, as a back-up, selected by the user.