Some diversity methods like Faith's PD and UniFrac require a tree, but there is no problem using the complex full tree without filtering. 'Full uncut' trees are fine for diversity analysis as you never see them. I only make trees less complex right before I graph them. 
The rest of the time, I use the full tree.
Normalization is very contentious. 
I'll say stick with the methods in the tutorials and see that reviewers suggest you do. 
Colin