What is the significance of the following sentence: "A rough, dough-faced, thoughtful ploughman emerged from a slough to walk through the streets of Scarborough, coughing and hiccoughing."?

This sentence illustrates the nine different ways to pronounce the syllable "ough". Answered from my own knowledge, backed up by various online sources, e.g.
https://www.englishclub.com/ref/esl/Power_of_7/7_Ways_to_Say_ough__2924.htm

Hang on, slough rhymes with plough or rough depending on meaning (or capital as in Slough - which rhymes with plough). That makes eight different ones in the question, not nine.

Also depending on your dialect of English, I'd guess...

Not so much. The actual IPA designation of the sounds will vary from dialect to dialect, but the rhyming is constant within each group. That is, rough never rhymes with dough, but always rhymes with tough no matter whether you say taff and raff or toof and roof - they don't ever match up with deow or doh.

Apart from (rhymes with plough), I thought slough was pronounced sloff (rhymes with toff) -- not sluff (rhymes with rough).

In the US it's pronounced "slue," rhymes with "glue," "blue," and. . ."through," leaving 8 pronunciations.
http://www.merriam-webster.com/dictionary/slough

A good list (two, actually):
http://everything2.com/title/English+pronunciations+of+%2527ough%2527

More detailed treatment:
https://en.wikipedia.org/wiki/Ough_%28orthography%29

The pronunciation sluff is for snakes getting undressed, and the rhyme with plough is the Town and the nasty boggy thing. (The town isn't quite that bad...) (Not quite - it's not boggy...) A point is that 'plough' is the Brit spelling (but it needs to be as plow isn't an -ough.) We do have a few words pronounced 'slue' - but they're spelled 'slew'. Which 'slough' is pronounced 'slue' - the bog or the snakey one?

The bog.

See Merriam-Webster above or click here:
http://dictionary.infoplease.com/slough