How should foreign words (with foreign characters) be written in English text?

Generally they should be written as they appear in their native language; this is the standard practice in the press, I believe.

The ideal would be to preserve the word as it appears in its native language, but it is something that English-speakers are very lazy about. Anything that looks orthographically odd (i.e. has glyphs that aren't part of everyday English) stands out and is usually thought of as pretentious, so there is a strong social pressure to normalize. Ligatures in particular tend to unlink into their component letters (think of ß as a ligature in this respect), and missing characters translate in various slightly inconsistent ways.

If a word becomes common currency in English, it gets normalized over time. In particular, the accents fall off: writing "café" is considered a bit affected these days, and "rôle" has pretty much died out, for example.

I've often seen phrases or words in foreign languages italicized in text to signal to the reader that the word is of foreign origin, and may therefore be later explained to the reader at the author's discretion. The author can include as many foreign characters as they feel is appropriate.

Be very careful about this, however. For example, if the reader has no familiarity with the language in question and the author decides not to explain the text in question, the reader may start to feel excluded or left out of the story.

One extreme example would be the teachers in Charlie Brown cartoons (not text, I know). Usually, the Charlie Brown characters will partially restate the trombone "wah-wo-whas" in their reply, so as to keep the intended audience included in the implied other half of the conversation.

The accents that are more likely to be kept in are the ones we're more familiar with. British kids are taught French so French words are almost never changed. Latin-based alphabets that the country is not familiar with, such as Scandinavian languages are more likely to be changed, either to french style or none.

The decision to transliterate or not depends on the original language. Russian, Sanskrit, and Ancient Greek are Indo-European languages that are usually transliterated. However, commentaries on original texts often do not transliterate. Some writers use both original and transliterated texts (Jacob Klein's A Commentary On Plato's Meno).

I am no expert in this matter, but as a native speaker of other languages beside English, I would like to contribute the following.

I have the impression that the OP is using the label "accents of all kinds" for things that fall in, at least, two very different categories: some are true accents, such as the one in "á", and some others are not, like the one on "ñ". The "á" in Spanish is still an "a" to all effects, but an accented one. However, an "ñ" is not an accented "n" but a totally different letter altogether. They are not exchangeable.

The Chicago Manual of Style (16^th edition), says that ligatures should be decomposed in Latin and transliterated Greek, as well as in words borrowed into the English lexicon. However, æ and œ should be used for Old English and French words respectively, when respectively in an Old English or French context.

There’s a whole chapter on foreign languages, but in general I’d preserve accents and “strange” letters when including words from foreign languages.

But it depends a lot on context as well. What kind of text are you writing, who is your audience? In an academic context the answer is usually pretty straightforward (just see how books and papers in relevant fields do it), but if you’re writing for a wider audience, simplifying may be prudent.