Relationship between kingdom, dominion, and doom

As your dictionary told you, the resemblance between the -dom suffix and the dom- prefix is coincidental. In European languages in general, affixes are either prefixes or suffixes, but very rarely both. I'm not aware of any affix at all in English that can be both a prefix and a suffix, so the fact that dom seems to occur in both places is a pretty big clue that the two roots are unrelated.

And dom- is not really a prefix. Rather, it's from Latin dominus "lord", which is in turn related to domus "home, house". Words that come directly from dominus include dominion, dominate, domain, while words coming from domus include domestic, domicile.

Now, the English suffix -dom comes from OE dom, which is a different word entirely meaning "state, condition, authority, jurisdiction", which despite its similarity in meaning and form is not related in any way to Latin domus. Rather, it comes from Proto-Germanic *dōmaz, from a stem verb originally meaning ‘to place, to set’

According to EtymOnline:

The -dom suffix comes from the Old English word dom (pronounced like dome), which meant statute or judgment. That is also where we get the word doom.

Dominion on the other hand ultimately derives from the Latin root dominus.

In Scandinavian languages, which share the same heritage as Old English, the term "dom" is still used in various words referring to judges, judgments, courthouses, etc. and also in older terms for various titles, such as bishoprics (bispedømmer) or kingdoms (kongedømmer).

I looked the etymology up because I figured these words were derived from the architectural term "dome", from Latin, which is also found as a prefix for the Danish word for cathedral, "domkirke". On further reflection, this makes little sense in many cases, and indeed turns out to be wrong; In Danish as well as in English, the term "dom" for domes/domiciles/dominion and the term "dom" for legal terms and various titles are similar due to coincidence. In Danish, to complicate matters, the terms can both be prefixes.

Ironically, I almost wrote that the Germanic term refers to "domains" like kingdoms - "domain" being another Latin word. It's easy to see why there's confusion.