1. Yes rabbits sleep. They are not nocturnal butthey sleep during the day too. Rabbits are crepuscular, which meansthey are active during the dawn and dusk hours.
2. Yes rabbits shut their eyes.
3. To groom during the periods they are shedding I use a "cat slicker"brush, it has bent bristles to get out the shed hair efficiently. Ifyou just want a brush that your bunny will really enjoy and increasethe bonding process, a soft bristled one is most enjoyable for them,but cat slicker brushes work wonders in removing hair more then thesoft bristle ones. As for the powders, I have no clue.
4. To bath them you can put them in a bath tub with a little bit ofwater, or a sink, or fill the bottom portion of the shower with waterand them that prevents them from being able to jump out. I would notsuggest bathing them if they don't like it! Rabbits are apt to freakout and very likely to cause damage to you or themselves. Rabbits needthe oils that they have and it's better not to bath them more thentwice a year as it is and it must always be dry before putting in acool place. Rabbits are naturally clean animals and don't need bathing,if their but gets dirty, simply wipe it with a wash cloth.
5. Rabbits bond best when you just sit and wait for them to come toyou, then you can pet them and allow for them to choose to come to you,if your bunny already does this, spend time brushing or pettinghim/her. Don't rush your rabbit and force it to spend time with you,instead allow it to choose to be around you, you can't force them to doanything! They will eventually find that if they come to you, they willget pats and they love attention.
Laura