I suggest that you learn the constructed language called Interslavic. In fact, it would be a service to the community if you learned it well, used it for communication, and then reported the results.
Interslavic is a language for communication between Slavs of different
nationalities, rooted in centuries of geographical proximity, shared
history and ...
The website Easy Croatian has 90 lessons (plus appendices) available for free (and for download) and seems very comprehensive - extensive & detailed lessons, and well-organized. However, it's not interactive like Duolingo is - easy-croatian.com is essentially a book.
Pimsleur also offers an audio course for Croatian, however, Pimsleur's method may not ...
Take the following sentence as an example that includes all your points:
Јаc сум ја прочитал книгата.
Jas sum ja pro=čita-l kniga-ta.
I1 am2 3-sing-fem-acc-clitic3 perf=read-past4 book-the5
'I read the book."
Clitic doubling: the presence of both the direct object книгата and the direct object pronoun ја.
As a native Czech speaker, I should be able to answer some of your questions, but not all. Usage of the cases is something I have learned naturally (without having any idea of grammar), so I can't help you there. It is also many years ago when I attended an elementary school. Sorry.
On an Android app: You might try Duolingo. I have some experience with that,...
Doubling of the Object- this probably refers to the use of two object pronouns in a sentence, specifically when describing the use of direct and indirect object pronouns with a verb.
Verb aspect (Perfective vs. Imperfective)- This probably describes the distinction between two different aspects for employing the past tense. In most languages, this ...