Assalaamu alaykum. I put my bedsheets/bedcovers in the washing machine due to the impurity of semen (I am aware that there are different opinions about this, but I take the more difficult view in order to be on the safe side - I am not sure if this is correct either). One of the bedcovers is waterproof (to protect the mattress). The washing machine was put on a three wash cycle (wash and spin three times). After the final wash, the waterproof bedcover came out dripping. The other two were wet/damp.
1. Are the bedcovers considered pure or impure?
2. The water that dripped from them onto the floor and the place where I put them to dry, have they been rendered impure?
3. Is the following true: If the impurity is thin (like urine and so on), then the cloth will become pure after having been washed thrice and squeezed all three times with all strength, i.e. no drop of water should drip from the cloth if squeezed once again. If one does not squeeze it with all his strength for the safety of the cloth, then it will not get pure. (Bahar-e-Shariyat, vol. 2, pp. 120). This is a translation of what I read on an internet site. Regarding question 3, I am not sure if this is always possible.
May Allaah reward you.

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.

Your question touched upon several issues, which we will answer in the following points:

1- The jurists differed in opinion about the purity of semen. Imaams Ash-Shaafi'i and Ahmad and those who agreed with their opinion held that semen is pure (Taahir), and this is the view that we adopt here at Islamweb. Imaams Maalik and Abu Haneefah held that it is impure (Najis) and that it has the same ruling as urine. Therefore, if one washes the clothes stained with semen in order to avoid the difference of opinion of those who consider it to be impure, then this is good. For more benefit, please refer to fataawa 85710and 84581.

2-  If we assume that semen is impure, then the bed covers that are stained with it become pure by washing them with water to the point that the water that runs down from them becomes unchanged by the impurity of the semen. Ad-Durr Ath-Thameen, authored by Shaykh Mayyaarah from the Maaliki School, reads, "The Ghusaalah (the water that runs down from washing impurity), if it is changed by the impurity, then it is impure, and if it is unchanged, then it is pure and its wetness that remains in the clothes does not harm as it is part of the Ghusaalah, so it is not an obligation to squeeze it out of the clothes."

3- If the water runs down unchanged by the semen, then this is enough. The water that drips from the covers after that, or while drying them, on the floor is considered pure. According to the more correct view, it is not an obligation to squeeze the clothes after the impurity disappears. Al-Kharashi said in his commentary on the Maaliki book Mukhtasar Khaleel, "The spot where there is impurity, if it is washed with pure water and the water runs off it pure, then it is not required to squeeze it; because the water left on the spot is the same as the water that runs off, which is pure.”

4- We have not come across the book that you mentioned 'Bahr Ash-Sharee'ah', but the statement that you attributed to it is the view of the Hanafi School regarding purifying the impurity that cannot be seen with the naked eye. This does not apply to semen, which is the issue in question. The Hanafi book Ad-Durr Al-Mukhtaar reads:

"The spot of the impurity that cannot be seen with the naked eye becomes pure based on the predominant opinion of the person who washes it believing that it has become pure without any specific number of times of washing as is the Fatwa in our school. For a person who is affected by Waswaas (obsessive whisperings), the estimate is to wash it and squeeze it three times or seven times for the clothes that can be squeezed tightly to the point that no water drips down. In case one does not squeeze tightly because the clothes are delicate, then do they become pure? The predominant opinion is yes; they becomes pure for the necessity."

However, Ibn Aabideen said in his commentary on Ad-Durr Al-Mukhtaar:

"It appears that squeezing tightly (with exaggeration) is a condition at all times, and, according to the book Ad-Durar, it is a condition in the third time only. But it should be a condition in each and every time as per the apparent view of the author of Al-Khaaniyyah, as he said, 'If one washed the clothes three times and squeezed it every time although he could do more than that, but he did not exaggerate in squeezing it as a way of preserving the clothes [i.e. fearing that the clothes will be damaged], then this is impermissible''" [abridged]

He also wrote, "His saying 'the most correct opinion is yes, it becomes pure for the necessity'; this is the same view as in the book entitled An-Nahr, quoting As-Sarraaj. That is to say, so that this does not necessitate wasting money. The author of Al-Bahr said, '...but the author of Al-Khaaniyyah had chosen the view that it does not become pure.'"

Finally, one should be very careful of being carried away with such questions as this may lead the person to being affected with Waswaas, which is a great evil that is difficult to get rid of.

Allaah knows best.