My last name/surname is West. Can I just write it as the kanji 西 nishi? It means West but it's also a last name. If not, how do you write West in katakana?
Depends on what you mean by "use".
In any legal or formal circumstances, absolutely no chance. Your family name will be 「ウエスト」 or 「ウェスト」 with a small 「ェ」.
If you ever become a Japanese citizen, however, you will be permitted to change your name from 「ウエスト」 to 「西」 if that is what you still desire then.
If you are just visiting Japan and want to use the family name 「西」 among your friends while there, then basically no one would stop you from doing that. It would be understood as a kind of humor more than anything. In that case, naturally, your foreign resident card will NOT say your name is 「西」.
"West" would usually be transcribed ウエスト or ウェスト (with the second kana the usual size or small).
I have seen people "translate" their surnames into Japanese, like
Silver → 白金【シルバー】
and print them onto business cards, but together with their actual name and with a transcription in katakana.
However, unless you become a Japanese citizen, you have no "Japanese name". Your 外国人登録証明書 (Alien Registration Card) will contain your name in rōmaji unless you can show that your legal name is written with kanji, for example if you are Chinese. (These kanji need to belong to this extensive list.)
Moreover, if you are a mid-to-long-term resident and you can show that you are being disadvantaged (e.g. discriminated against) because of your rōmaji name, the Minister of Justice may approve of replacing the rōmaji with appropriate kanji or kana.
In both cases the burden of proof lies with you, so getting a Japanese name is certainly not automatic.