Let's look at domeh first. The root is דמה, meaning "to be like, resemble, to be equal in value." The verb's meaning progressed from "likened" to "compared" to "considered" to "imagined". From this root, with the various meanings, we get quite a few common words, including:
- demut דמות - in Biblical Hebrew it meant "likeness" or "image". In modern Hebrew it primarily means "personality."
- dimyon דמיון - It only appears once in Tanach (Tehilim 17:12), with a similar meaning to demut, with the meaning "similarity", which it still has today. In modern Hebrew it also has the meaning "imagination" - perhaps in a similar way that "image" and "imagination" are related in English.
- tadmit תדמית - This word means "image" or "perception", particularly how one is perceived by others.
- demai דמאי - This is a halachic term for "produce not certainly tithed".While there are a number of folk etymologies for the word, Klein derives it from our root דמה and says it literally means "seeming, apparent".
- damim דמים - Klein says this post-biblical word meaning "money, value, price" is probably derived from the root דמה meaning "to be like", in the sense "to be equal". (There are, however, many drashot that connect damim as money to dam דם - "blood").
This root, דמם, "to be or grow dumb or silent" has cognates in many Semitic languages, such as Ugaritic, Aramaic, Arabic and Ethiopian.
You might have noticed that the English word "dumb", originally meaning "silent" has a similar sound to the Hebrew root. However, they are not related. All research I could find says that the English word "dumb" comes from the Indo-European root *dhumbh, which is (as Klein writes) "a nasalized form of base *dhubh or *dheubh, 'to fill with smoke, to cloud darken; to be dumb, dull, or deaf.". Cognate words in English may include deaf, dove, typhus and stove.
Now, I know that the Hebrew and English words sound similar. And they both have similar meanings - both "mute" and "confused". But while this is a good example of using your dimyon, they don't have the same roots. Remember the helpful Hebrew phrase דומה אך שונה - "similar, yet different."