As for taking eucharists home: yes, they did that early on. But it became problematic as people (I am told) began to invent superstitions about it, and otherwise abused the privilege. Much like public confession of sins, the practice went away.
You can still see vestiges of this today though. For example, Antiochians still take some of the Holy Unction home with them for future use. This does not happen among Russians. For whatever reason, the practice was never prohibited in Antioch and it still survives.
An example of superstition, about Unction, would be anointing clothes with it for good luck or something, which (I am told) happens in Romania, to the chagrin of the bishops. In ancient times, Christians may have put the Eucharist bread crumbs around their house to protect from demons, say.
This kind of misuse forces tighter restrictions on the administration of sacraments.