if the second want be right that should like :他喜欢躺在床上看书，而不是他喜欢躺着床上看书

The third one is correct.

1）他喜欢看书躺在床上，眼睛越来越不好了。 ==>if the above sentence change to this one ex: 他寫歡看書"並且"躺在床上 , "所以"眼睛越來越不好了。 2）他喜欢躺"在"床上看书， "所以"眼睛越来越不好了。==>"著" change to "在" 3）他喜欢在床上躺着看书， "所以"眼睛越来越不好了。(right) I hope to add all the sentences with "所以" behind the "," . it represent the result . Actually ,(3) is right . If you talk to Chinese people (1) ,(2) and (3) , they can still understand what you mean. I hope that can help you in Chinese .

2nd and 3rd both seem right to me.

3 is totally correct if 2 is "他喜欢躺在床上看书", it would be also correct. "着" means a state.