Learn English – How to say just want to let you know politely

There's nothing at all wrong with the "I want to make sure that" part of your sentence; the part that I find a bit obnoxious from a politeness aspect is the passive construction "it would be appreciated." You'd be much better off expressing your appreciation directly rather than simply allowing for the possibility that some unknown object somewhere will be gratified:

Further, I'd like to make sure that the dates of the courses will not conflict with my other appointments, so I'd be grateful if you could supply more details about the arrangements and what their costs would be.

(There are also a few other minor things that I've tweaked, particularly the placement of "that" in the first phrase, and the use of the singular pronoun "it" to refer back to the plural "arrangements".)

There's a bit of a disconnect in that you've specified that you care only about the dates but then you go on to explicitly request cost information as well; but if you've mentioned the cost aspect prior to this sentence, it's not a problem.

In the UK, black person is the usual way to describe someone of African or Caribbean ethnic background and I wouldn't expect it to be taken as offensive. Referring to someone as a black (as a noun) would be offensive.

Referring to someone as the black guy could conceivably be interpreted as a little disrespectful if you might have been expected to call them by name, depending on the context. In your specific example you could have said I don't remember your colleague's name but he's black, if that helps? and I wouldn't expect anyone to be upset by that form of words.

Your friend is either misinformed or engaging in propaganda against perceived "political correctness". Stories about the word "black" being banned in some context or other pop up in the tabloid press with depressing regularity but invariably turn out to be untrue or misreported.