So then typing Japanese on a keyboard still takes several key-presses for each word? How does writing speed compare between English and Japanese on a keyboard?

Furthermore, how was the placement of the characters decided? Are they more closely related to QWERTY (so that typewriters don't jam) or to Dvorak (so that the most frequently used letters are on the home row, and so that alteration between left and right hand is maximized), or unlike either? I use Dvorak and if I were to learn Japanese and type it on a keyboard, I'd want the typing experience to be similar to how it feels to type on Dvorak for English compared to QWERTY.

Yeah, it takes several keypresses per word. Speed is pretty comparable to English as the process is very similar to spelling English words.

To explain how it works in detail, it's important to first note that Japanese doesn't really use spaces between words. This probably sounds weird if you're used to English / most Western languages like it'd be hard to read but actually it's not. You basically compose one word at a time and it draws an underline under the word you are currently typing, then when you press space bar it autocompletes to the correct ideographic spellings. Pressing space again let's you cycle through different possible transliterations (including leaving it spelled in phonetic characters); it almost always gets it right except for homophones (which there are a lot of in Japanese) or if you want to deliberately pick some unusual/archaic spelling for stylistic reasons. Then you proceed to composing the next word. Sometimes dedicated Japanese keyboards have a separate button for the "autocomplete" function, but most use space bar AFAIK. Wikipedia has a description with a demo image of the Windows IME (they all work pretty similarly AFAIK) https://en.wikipedia.org/wiki/Japanese_input_methods . Cell phone input described on that page is where things get interesting / deviate more if you're interested.

So there's a small amount of extra overhead with selecting the correct transliteration, but it's minute once you get used to it. Japanese is I'd say slightly more information dense than English, so it compensates and the typing speed is about the same. I'm not actually sure how the character layout was chosen for the Japanese keyboards. Looking at it roughly, I'd guess that the placements are approximately matching the English usage frequency of QWERTY corresponding to the frequency of usage in Japanese as that looks about right to my eyes, but that's just a guess.

You can use Dvorak or whatever you want to though. You don't need a specialized Japanese keyboard. Instead of typing the kana (phonetic characters) directly you just type their Romanized form. So instead of typing たべます (phonetic spelling of "I eat") you'd type "tabemasu", but it'd otherwise behave as I described. I know a lot of Japanese that don't even bother using proper Japanese keyboards and just use standard English keyboards, especially programmers. You'll have to fiddle with some settings, but I know for sure that it can at least be done on Linux and Mac and I'm sure Windows can do it too.

Edit: to explain how Japanese writing works a little better, there are actually 3 "alphabets" - two phonetic (hiragana and katakana) plus the ideographic alphabets, called kanji. The phonetic alphabets always correspond to the same sounds, whereas the kanji refers more to a meaning/idea and can can be read to correspond to multiple different sounds depending on context. For example 食, the character for eating/food is pronounced "ta" in "taberu/食べる" ("to eat"/"I eat"), but "shoku" in "shokudou/食堂" ("cafeteria"; the two characters literally mean "eating room").

Kanji usually only serve as the "root" of a lot of words and Japanese writing tends to be a mixture of ideograms and phonetics. For example, 食べる, where べ ("be") and る ("ru") are phonetic characters. If you conjugate the verb to for instance past tense 食べた ("tabeta"/"I ate"), the root character 食 that means "eat" stays the same but the phonetic characters change to indicate tense. It's also completely valid to not use kanji and spell things out entirely phonetically and this is how most people learn starting out this way and gradually replacing them more and more kanji as they learn, but to do so is considered childish / uneducated.

Japanese is a lovely language, I recommend learning some. Relative to English I'd say it's actually grammatically much more organized / logical (and hence easier), but on the other hand reading and writing are significantly harder to learn. The easy availability of manga/anime/novels in both untranslated and translated forms across a wide range of language levels makes it much more accessible than it was even just 10-20 years ago.