Repeating tasks (option 2) is always silly because it is inefficient.

I see no reason to merge things (option 3).

Option 1, in an extended form is what you want. While in essence from a task or project management point of view home and work are the same thing, it is entirely reasonable and practical to separate the two. You manage them in the same way, but you have two completely separate processes. So option 1 is entirely reasonable.

When you have information floating around in your head that you want to write down, yes you now have two places to go, but it is very clear which one you should write it down in.

When you're at work you will only look at Asana and not your home system. But if you have a thought that relates to home, put that down in your home inbox.

Your emails obviously should be separate too. Everything about work is separate for this to work, because that makes tasks and information very clear at all times as to which place they should go into. If there's a blur between home and work then the system might fail because you'll end up with a situation where you don't know where a task or piece of information is - home or work. That's why this way needs a clear separation.

As to using asana for gtd - you can use tags for next actions and/or other things, or you can use the 'my tasks' area as the inbox, next (today), later, follow up. I use an entire project for my inbox, and add random tasks into it, before moving them to right place when I do that. On my mobile phone I have that project as a widget so I can add tasks very quickly.