"Do you use a chair?"
I wonder why should someone use kanji when no of letters are not getting reduced? Here Hiragana and kanji both use 2 letters to represent chair.
Kanji isn't about reducing the number of letters (in fact, it usually takes longer to physically write something in kanji). Hiragana simply conveys pronunciation, while kanji conveys meaning. Kanji also makes sentences easier to read by making it clearer where one word ends and the next begins.
No. "Will you use the chair?" Essentially "Are you going to sit down?". Nothing about wanting to do something. It's not the volitional form here
No, for this you would use the -tai form , i think "chikaitai" would be the right form.
You don't use -tai form to ask about other people's wishes in formal Japanese, it is considered intrusive and impolite. They would instead use a different more elaborate construction to specifically avoid that.