It is true. These languages are all historically related, being part of a language family known as Indo-European, and those words are representatives of the root reconstructed as *dem, meaning 'house, household'. Briefly, most languages of Europe (excluding Basque, Uralic languages, Turkish), as well as those of Iran to India (and neighboring areas), are members of this language family. German, Norwegian, English, Icelandic etc. for one branch (Germanic); Hindi, Panjabi and so on descended from Sanskrit are another branch (Indic). This web page at Texas provides a wealth of information on lexicon for further study.
"Domus" and "domain" are via Latin – English borrowed lots of words. The Slavic words, on the other hand, are indigenous Slavic inheritances. A similar common word is the root for father, *pəter. The correspondence English f: Latin p reflects a sound change (Grimm's Law) in the history of Germanic, and that (among other things) is how we know that English domain comes via Romance and Latin.