Hi
I just quickly answer for your questions.

For the first question,
“I just asked myself a question: I remember that the Japanese have kanji (and other characters which are combined with them (hiragana?!)).”

Basically, we have three different character types and also we often use English alphabets characters.

• Three ideograms using the Japanese script.
• All are used together when writing.
• Most phrases are a mixture of hiragana and kanji.

1.HIRAGANA　ひらがな.

This ideogram is used for non-Japanese words and names or new words tend to be in this alphabet.
2.KATAKANA 　カタカナ.

Similar to Chinese characters.
3.KANJI 漢字.

Plus ALPHABET


For the second question
“However, if I remember correctly japanese can be written in our latin alphabet, too..I've heard of people learning to speak japanese (or chinese) without learning all the kanji (because they only needed to be able to speak it, not read it).”

I am not sure where this information comes from.
We learn all characters for reading, writing and for speaking as well.
Especially, Kanji once we learnt how to write, we have to know two different ways of how to read, it is one of the most complicated things for us too. Historically it is further more completed so, I will explain for current usage only. Kanji
character is shared by two way of reading. One is call “On-Yomi(Chinese-reading)” which is used for only Kanji reading. In case, it is not always as the same as real Chinese speaker’s reading, now more developed as Japanese way)
Then, also we have ”Kun-Yomi (Japanese-reading)” which is used with Hiragana.
I am not sure about “Latin alphabet” you mentioned. We do not have learning like Latin language in UK. If you meant this, we don’t have similar type of use.

The third question
“I'm not sure about all this (it has been some time since I was interested in learning an exotic language), but do japanese people ever search for keywords using latin characters?”
We use English style characters for if this name, phrase or word is very popular, but it is unlikely to be used as search term than term searched in Japanese characters. Even same reading sound both English and Japanese but a term written in Japanese is more likely to search.

The fourth question
“So for example a competitive keyword might have a ton of searches and a ton of competition - if searched for in kanji, but might be less competitive (and of course a way smaller search volume) if it's spelled in latin characters.”
If you are targeting Japanese people who use Japanese, I suggest that you would use more Japanese character keyword than English characters or other language.
And then, when use only Kanji for search, sometimes the result will show includes Chinese language people’s website. It happened to my search when I use Google bar search box or when I forgot to search from Japanese SE page.



The fifth question
“Does this ever occur? Are there japanese websites spelled in latin characters?”

If you go to the company site of Honda, you will see some English phrase but those are more likely to use as design than the main purpose of web contents.
Example: “Honda”.