Hey Ken, great textbook you're using. I also used the Genki textbooks to learn Japanese. Finished 1 and 2. I found it a bit annoying how the Kanji's introduced in the grammar chapters were not synonymous with the kanji's being introduced in the corresponding Kanji chapter at the end of the book. My advice to you would be to use Genki mostly as a grammar/vocabulary textbook. In short, you should memorize all of the Kanji's in the Genki textbook, but as you are probably starting to notice, this is way easier said than done. I'd recommend downloading a flash card application on your phone (I use Memrise). This way you can upload the kanji's you'd like to memorize and then practice them on the go. I think your strategy of doing the exercises in Kana is fine, however you should notice which words often come up and try to memorize the Kanji for those words. There are lists available on the internet too, of the top 100/500/1000 etc. used kanji. Those are helpful too. Good luck with your studies and if you have any questions please don't hesitate to message me!

You don't have to do them at once. I don't know about a text book you're using but I am writing as a Japanese. Looking back my childhood, I wrote all sentence in hiragana or katakana when I was a little kid. then I gradually started picking up kanji after I acquired basic grammar. I think it is quite natural as a learning process. Kanji is difficult even for native Japanese sometime. So don't worry, keep up what you're doing. Continuation is a key to success.