Translation:I normally have my breakfast here.
When there are differences of opinion, a dictionary may help. Larousse offers that petit-déjeuner is a verb meaning "to have breakfast" while petit déjeuner is a noun (m) meaning "breakfast". How does that square with your view? Apparently, this difference has not been adequately addressed in the examples used by Duolingo.
It depends on where you're from.
In Australia, we use "breakfast" as a noun (I eat breakfast here), and hardly ever as a verb ("I breakfast here).
I am aware of a trend of "verbalising" nouns, so am unsure if this is another example of this phenomenon.
(Eg. "friend". "I made a new friend on Facebook whose name is Sam" or "I befriended Sam on Facebook" has now become "I friended Sam on Facebook". )
In regards to your second question, it is similar to the English definition of "take". In English at least "I take breakfast here" means "I eat breakfast here".
"I eat breakfast normally here" sounds very idiosyncratic and maybe a bit childlike as well; I could easily imagine kids saying it like that, usually the ones who haven't fully mastered English rhythms. Said like that, it sounds virtually breathless and abrupt, like their speaking and thinking speeds aren't quite in sync yet. Why yes, I am an elementary school teacher!
Another popular option: "Normally, I eat breakfast here," which sort of implies that right now you are NOT eating breakfast here. And as camembert points out, I'd never say, "I eat breakfast normally here," unless my friend and I were sitting at my favorite restaurant and he recommended that we eat breakfast while standing on our heads. "No, no, I eat breakfast normally here."