Is there a prohibition against bringing children to a shiva house? I know that children aren't usually brought and that it's respectful to ask in each case, but does anyone know if there is an actual halachic prohibition against this?
This may be a bit of a stretch, but consider Shulchan Arukh YD 391:1
אבל אסור בשמחה; לפיכך לא יקח תנוק בחיקו כל שבעה, שמא יבא לידי שחוק.
A mourner is prohibited in "happiness"; therefore he should not take a baby in his bosom all seven days lest he come to playfulness.
This seems like an "actual halacha". The age of the "תנוק baby" is not specified anywhere as far as I can tell, though it seems reasonable based on the reasoning that a toddler that you ordinarily might plop on your lap and play with is included.
This rule is not about entering a Shiva house, but if there is such a custom it likely derives from this rule about holding young children and a general sense of minimizing the happiness which children can create.