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.
It is permissible to hang the picture of the Ka'bah or the picture of its door. It is also permissible to hang the frame on which the Names of Allaah or the name of His Prophet are written. However, if one intends by hanging them to get blessings from them or to repel a harm, then in this case they are included among the forbidden amulets according to the view of many scholars and the view of the companions among whom are Ibn Mas'ood and Ibn 'Abbaas .
As regards performing the prayer in front of the frame of the door of the Ka'bah, then it is a condition for the prayer to be valid that it is performed to the direction of Makkah. So, if the frame on which the door of the Ka'bah is designed is in the direction of Qiblah (the direction of Ka’bah), then the prayer is valid. However, seeking to perform the prayer towards the picture of the door of Ka'bah could be classified as an innovation, so you should avoid it. As a matter of fact, it is confirmed that the Prophet said: "Whoever introduces into this matter of ours [Islam] that which is not a part of it, it will be rejected." [Al-Bukhari and Muslim]
Furthermore, seeking to perform the prayer in the direction of the frame of the door of Ka'bah may lead a person to believe that he gets more reward or attains other merits by doing so.
Concerning putting this frame or that which contains the Beautiful Names of Allaah in the bedroom, please refer to Fatwa 90983.
For more benefit, you may also refer to Fatwa 83322.
Allaah Knows best.