When/How often should I start writing the kanji down

On the ‘reading all kinds of fonts’ part, you can try installing Jitai (字体): The font randomizer that fits on WK. Also, you can try changing your browser’s default font for Japanese, as well. I’ve recently set it so it shows up like this:

For handwriting practice you can look up ペン習字 on Youtube for videos that give tips on how to write pretty. I like 書道家 東宮たくみ:

There are apps as well to help you oractice strokeorder, or you can look up stroke order on jisho, or add it to your lesson routine using this script: [Userscript] An Unobtrusive Kanji Stroke Order, or this script: [Userscript] Stroke Order Diagram