Is it unusual in a noticeable way to slant your Japanese writing? As a right-handed person whose first language is English, I have a natural rightwards slant to my handwriting much of the time, depending on how quickly I'm writing and what physical position I'm in as I write. I've noticed I sometimes impart this slant to my Japanese writing as well, when I'm practicing writing out sentences. Is this as normal in Japanese as it is in languages that use the Latin alphabet? Will rightwards-slanting kanji/kana seem odd? I haven't been involved with Japanese long enough yet to even learn whether there's such a thing as italics in Japanese, so I'm not sure how it would be perceived.

In general, writing slightly tilted _horizontal_ lines is considered beautiful. This link has some beautiful handwritten sentences. You can see many horizontal lines are sloping slightly upward to the right. The faster you write, the more slanted lines will be, just as you write something by hand in English.

Of course you should not overdo it. A few Japanese people have habits of writing overly slanted characters. Here are some examples I found on the net:

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Actually these are not necessarily dirty, and some may think they are rather good.

On the other hand, vertical lines should generally be completely vertical. But many people have habits of writing slanted vertical lines to varying degrees, too. If your handwritten characters are like this, I would say you should eventually correct your writing.

There is no such a thing as italics in Japanese orthography. If you want to emphasize some words, the first choice is to underline them.