Just don't get mulch right up against the tree, if you are going to mulch the area. The flowers will not have any affect of the tree's ability to get water and nutrients, since most of that occurs at or near the drip line where the edge of the branches hang. The only problem you might have is what I have with my trees, and that is that there is not enough water that gets through to the ground under the trees, and plants languish. If it is too shady, very little will grow well (another of my problems).....even hostas do not do well for me under my trees.
If you are new at planning gardens, you will probably make some mistakes, as we all do. Just don't be afraid to undo things that are not working the way you want. Take that advice from someone who hasn't done that, and now has regrets. You also might try something like just burying some pots of mums around the tree in the pots to see how you like the looks of it before you commit completely.
I am envious of you having a huge elm tree....how did it survive the Dutch Elm Disease?