Is the following a proper sentence?
His sincerity concerning intimate issues made the old man increasingly uncomfortable
I am not sure about using 'concerning' with sincerity
I'm not sure whether your question more concerns the use of 'sincerity' or the use of 'concerning', or proper sentence structure. However, per the title of the question, I would say that sincerity is not the best word to describe openness in this context.
Sincerity indicates honesty and straightforwardness, but does not necessarily imply that one will offer the truth without being asked. I believe you may be looking for the word candor. Per MW's definition #4:
unreserved, honest, or sincere expression : forthrightness
Specifically, candor's connotation of forthrightness seems to better fit your example sentence. A merely sincere person may not necessarily be forthright enough to bring up a topic that makes others uncomfortable.
Yes, it's a proper sentence. Concerning, regarding, etc. would be fine in that sentence. Concerning is properly used as a preposition in the noun/prepositional phrase form:
noun + prepositional phrase