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 is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
As regards putting the Quran on the floor, if there is anything on which the Quran could be put or that it is not difficult for someone to hold it up with his hands when reading it, then there is no doubt that holding it up is a must, as this is honouring it. It is confirmed that some Jews called the Prophet to the place where they used to study the Torah and asked him: "O Abul-Qaasim, a man amongst us has committed adultery (or fornication), so judge between us.'' They put a rug for the Prophet and he sat on it. Then he asked them: ''Bring the Torah to me.'' So when they brought the Torah to him, he took the rug from underneath him and put the Torah on it. [Abu Daawood] So the Prophet put the Torah on the rug as a sign of honouring it, so the Quran deserves more honouring and respect. The scholars mentioned that it is forbidden to put the Quran on the floor. Al-Bujayrami [from the Shaafi'ee school] said: 'It is forbidden to put the Quran on the floor, one should lift it up even a little bit from the level of the floor.'
So you are right in denying and complaining why the sisters are putting the Quran on the floor. However, if there is nothing on which the Quran can be put, and there is a difficulty for the reader to hold it up in his hands, then putting it on the carpet in front of him is not considered as disrespecting the Quran provided the carpet is clean and pure.
With regard to a woman covering her head when reading the Quran, she is not obliged to do so unless she is in the presence of marriageable men in which case she is obliged to cover her head and wear the proper Hijab. As regards doing so just for the sake of reading the Quran then this is not an obligation and we do not know any evidence which suggests that this is an obligation, and we have never heard any scholar who said that this is an obligation. Nonetheless, if a woman does so out of respect to the Book of Allaah, then it is hoped that she will be rewarded for doing so. Allaah Says (which means): {And whosoever honours the Symbols of Allaah, then it is truly from the piety of the heart.}[Quran 22:32].
It is reported that when Imaam Maalik wanted to relate the Prophetic narration, he used to wear his best clothes, apply perfume and sits in a respectful and calm way. However, one should not make a difference in this case between reading the Quran from the book and reading it from memory.
Allaah Knows best.