Interesting. It is not my area but everything I have read to date gave Halifax and Churchill as front runners, but not as the only possible options.Hastings and Beevor both definitively rule out Eden as a possibility on the grounds that the pro-war faction's support would certainly not have bled into him; their continued support for Churchill was about as based on personalist loyalty to the man himself as much as policy similarities, which Eden distinctively lacked at the time. Every author that I've read so far on this subject matter clearly reiterate that any alternatives to Churchill in the pro-war caucus were severely lacking, and that the bulk of Churchill's initial support was due to Chamberlain's continued support of him throughout the war cabinet crisis, which he would almost certainly have given to Halifax had Churchill been unavailable.
Hastings and Beevor continue on to talk about just how touch and go the War Cabinet Crisis actually was.
If, as you say, Halifax is the only option, then he is far from an ideal one. He cannot form a national government as Labour will not back him. He will also need someone to lead the Conservatives from the Commons while he sits in the Lords. It isolates him somewhat. A war cabinet would help put him at the centre of things, but it is still not ideal.
Possibly, but as noted above, very far from guaranteed. Battleships have torpedo defence systems so that they can (in theory) take a few hits before going down. Unlike with Royal Oak at Scapa, U-56 will not get a second shot at Nelson. That means the first salvo has to put Nelson down. It’s possible but not the most likely option. U-56 would have to be even more lucky than she was IOTL.U-56 in more able hands than Zahn could possibly have sunk the HMS Nelson and killed both Churchill and Pound, either on the 30th or the 31st.