I'm assuming there's a masculine form of model but I'm unsure what it is, any help?
Is this word related to "example"?
להדגים = to demonstrate
Cool! This is my favorite thing about Hebrew - once you know some words it's easy to know related words even if you never formally learn them.
Has the word anything to do with דג?
Probably not. דג seems ancient semitic (appears in Ugaritic), while דוגמא, who yielded the root דגמ, was loaned from Greek.
Example = דוגמה
Why does it sound like צמלה?
It's probably because of the 't' sound from לובשת: "loveshet-simla". When you speak fast, 't' and 's' sound like 'צ.
(Even if you speak slowly they combine. Only if you take a very artificial breath between the words there is no צ.)
Why is "The model wears a lovely dress" wrong?
Morphix Hebrew dictionary spells this word דֻּגְמָן. Which is correct?
Both. דוגמן is a male model and דוגמנית is a female model.
Or do you mean with or without vav? Again, both. When you write it with nikud, it will be without vav, and when hou write it without nikud, it will be with vav.