Is it obligatory to perform ablution (wudu') if rabbit blood gets on the hand, or is it sufficient to wash off the blood? And is it permissible to skin a rabbit while it is in the throes of death to facilitate its skinning?

Ablution is not invalidated by blood or impurities falling on the one performing ablution; it is sufficient to cleanse them. It is disliked to skin the slaughtered animal before it cools, due to the increased pain it causes, but this does not render the slaughtering unlawful.