What is the Sunnah as regards entering and leaving one's own house? Should we go out with the right or the leg foot first? Is this supported by any hadith of Prophet, sallallaahu 'alayhi was sallam? We know that there is a supplication at the time of sleeping: Allahumma (Bismika/Bi-ismika) amutu wa ahia (O Allah, in Your name do we die and live). I am a little bit confused about the correct pronunciation of 'Bismika' or 'Bi-ismika', which one is correct?

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.

Part of the Sunnah of entering or leaving one’s house is supplicating. We have mentioned the reported supplication of entering one’s house in fatwa 278833; so please refer to it.

As for the supplication of leaving the house, there are authentic wordings that were reported from the Prophet, sallallaahu 'alayhi was sallam, including the following:

1) Umm Salamah reported, "Whenever the Prophet left my house, he would raise his eyes to the sky and say, 'Allaahumma inni a‘oothu bika an adhilla aw udhall, aw azill aw uzall, aw athlim aw uthlam, aw ajhal aw yujhal ‘alayy.'" (O my Lord, I seek refuge in You from falling into error or going astray, or wrongdoing (others) or being wronged, and from behaving or being treated in an ignorant manner.) [Abu Daawood, An-Nasaa'i, At-Tirmithi, and Ibn Maajah; this is the wording of Abu Daawood]

2) Anas ibn Maalik narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said, "When a man goes out of his house and says, 'Bismillaahi tawakkaltu ‘ala Allaah wala hawla wala quwwata illa billaah,' (In the name of Allaah, I place my trust in Allaah; there is no power and no strength save in Allaah.) then the following is said to him, 'You are guided, defended and protected.' So the devils will move out of his way, and another devil will say, 'How can you deal with a man who has been guided, defended, and protected (by Allaah)?'"[Abu Daawood - Al-Albaani graded it Saheeh (good)]

As for entering or leaving the house with the right foot first, the Maaliki scholar Khaleel ibn Is-haaq, regarding the etiquette of entering the bathroom, said, "It is recommended to enter with the left foot and leave with the right foot, unlike entering and leaving the mosque and house." There is no specific evidence for this; rather, it is based on the general statement of ‘Aa’ishah who said, "The Prophet, sallallaahu ‘alayhi wa sallam, loved to start with the right side in all his affairs; in wearing his sandals, in combing (his hair) and in performing ablution." [Al-Bukhari and Muslim]

As for the correct pronunciation of the word that you asked about, it is to say 'Bismika', and not 'Bi-ismika'.

Allaah Knows best.