Is there any way that I can find the grammar rules, that are sometimes explained with the new words? I think that would be really helpful to have all those explanations, tables etc. in one place, as we have the word list.
You're describing a basic grammar book (except for specific linkage to new Duolingo words). There are tons of them online and in hardcopy. Here's a start: http://www.hertie-school.org/fileadmin/images/Downloads/Student_Affairs___Study_Abroad/Online_language_learning_resources.pdf
As macjory says, a good grammar book is cheap and indispensable. Also if you open a skill you can see what grammar is covered by the skill just beneath the lessons.