Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
Does an internal examination require ghusl?
Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.
What should a person who was favored from Allah with a newborn, but couldn`t afford an Aqeeqah, do?
Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth) is a desirable Sunnah for the financially able since Allah, The Exalted, charges not a soul beyond its capacity. Therefore, if the father couldn`t afford the Aqeeqah before the end of his wife`s confinement, then it isn`t due on him, and if he was able to afford it later on, then it is permissible, but if he didn`t until the child reached puberty, the latter can offer the Aqeeqah himself.
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.