Should I tell my recommendation writers about my self-studying since graduating? Should I also mention my illness and depression?
First, I agree completely with Arno's answer here. The logic from that answer applies to these questions as well. Specifically...

Should I email the professors which will be sending letter of recommendation [and tell them] what I have been doing since I completed my masters in June 2020? (It would only list topics I have studied till now)?

Those reviewing your application may be interested to know what you have been up to during the past year. The answer appears to be &quot;studying.&quot; So, it may be worth listing this briefly in your application somewhere. Just one bullet on your CV, or one sentence in one of your essays.
To your question: I see no harm in also providing this information to your letter writers. For the most part, their letters will focus on what they know about you directly, not what you say you did after graduating. But, it's definitely a good idea to stay in touch with your letter writers, and mentioning your recent activities is a natural thing to say in this context.

[Should I] include episodes of depression along with stomach infection?

No. It's usually better to declare victory rather than make excuses. In your case, there is no need for excuses; it is quite straightforward to say &quot;since graduating, I've been studying topics X, Y, and Z.&quot; This sounds much better than &quot;I've been trying to study, but between depression and illness I haven't made much progress.&quot; Why volunteer these extra details?

[Should I include] the reason behind the depression [issues with my father]?

Definitely not. This is very much oversharing. It is understandable that you want to discuss your difficult circumstances and are seeking forgiveness, validation, and/or reassurance. But you should have this discussion with your family, friends, or therapist, not with your professional references.