Don't know if domu "comes from" Latin or some other common language, but "domus" means home in Latin. I know of derivative words in Italian and English (e.g., I am domiciled in the state of Virginia = I am a legal resident of the state of Virginia).
I remember hearing that Polish has a lot of Latin / Italian words (particularly for food) because several Polish queens came from what we now call Italy. E.g., ananas in Polish = annanas in Italian = pineapple.
Polish has many latin loanwors like most European languages, and many food related Italian loanwords from Queen Bona , but "dom" is older than those.
Polish linguists at PAN say it has proto-slavic roots ( *domъ), which means it comes from times before slavic languages split.
I do not know what @Vegnir's source is but according to him both latin and slavic come from their ancestor Proto-Indo-European *dṓm
In English, "we are going home", is not the same as " let's go home".
The first is telling someone what you ar 'currently' doing (or, strangely, what you are going to do after finishing the sentence).
The second is telling someone what to do with you. It is actually used as an order to be followed, but ordering/suggesting/asking quite politely. ('Let us' has many different meanings. It can mean a request to allow you to do something, but its also a suggestion when amongst friends. Eg "Let us consider this idea")
If "lets go home" is said abruplty, angrily, then it could be slightly similar to "we are going home". Meaning you are indicating EXACTLY what is about to happen next.
All a bit hard to explain in writing