All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
There is no doubt that respecting and venerating the Quran is an obligation and is considered part of glorifying Allaah The Exalted Himself, since the Quran is the Speech of Allaah. Imaam An-Nawawi said in Al-Majmoo’: “The Ummah (the Muslim nation) has unanimously agreed upon the obligation of glorifying the Quran in an absolute sense by exalting it and preserving it.”
Shaykh Al-Islam Ibn Taymiyyah said in Majmoo' Al-Fataawa: “Respecting the Quran is obligatory wherever it is written, just as writing it is forbidden wherever it might be debased by the urine of animals or like someone sitting on it (etc.) according to Ijmaa’ [complete agreement of Muslims].”
Nevertheless, there is nothing wrong with sitting on a chair while someone else is seated on the floor reading the Quran. This is neither a form of debasement of the Quran nor a form of disrespect. ‘Aa’ishah said: “The Prophet, sallallaahu ‘alayhi wa sallam, would recite the Quran while his head was in my lap and I was menstruating.” [Al-Bukhari and Muslim]
Ahmad related in his Musnad the Hadeeth of Maymoonah that the Prophet, sallallaahu ‘alayhi wa sallam, would enter upon her while she was sitting during her menses, so he would recline on her lap and recite the Quran.
This is evidence, as his recitation of the Quran while his head was in her lap is not considered debasing the Quran, so the same applies to her sitting up as he would recite while reclined. There is no debasement of the Quran in that.
On the other hand, what the questioner mentioned about the way in which one respects the Quran by acting in accordance with it and by reciting it, this is true. However, respecting the Quran and venerating it are not restricted to this alone, rather one must also avoid all forms of degradation to it, like reciting it in places where such recitation is forbidden, like the toilet and so forth.
Allaah Knows best.