(～する)ことかなわず is a stiff set phrase that roughly means the same thing as (～する)ことはできず. This かなう means "(for a wish) to come true". は is omitted in this set phrase probably because it's an old expression.
My house is poor, so I can never welcome you into my house, ...
なにぶん used with a reason is a kind of intensifier (perhaps "at any rate" or "after all" is the closest?). 身の上 by itself refers to one's personal history/circumstances, not one's body. This ～にて is an archaic way of saying ～であって/～でして ("is ～, and/so ..."). Xは容赦してください (or Xは許してください, Xは勘弁してください) literally means "Please be forgiving concerning X". It's used when X is the last thing one wants to do, so it's like "Please, no X!" or "Please, I can do anything but X".
(After all,) I have been sickly (for a long time), so please forgive me for not coming under the sun.