Politeness – Is It OK to Respond to ‘Thanks’ with ‘Sure’?

Yes, it's rude. "Whatever" expresses indifference; often, expressing indifference is dismissive, and in this case, it's dismissive of what the other person has to say. Semantically, it's equivalent to responding with "I don't care".

Being dismissive is what makes it rude. Since "whatever" can express indifference without being dismissive, it's not rude in all situations. If a friend asked me what I wanted to eat, and I replied "whatever" in a non-sarcastic tone of voice, it probably wouldn't be taken as rude. Rather, it would express that I had no particular opinion and that I'd eat whatever they wanted to eat.

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.