So, assuming, that Representative Minnick of Idaho (just for example) is places as a designated survivor and he's only one left, he can elect himself as a Speaker and assume the Presidency, correct?
Possibly. The controlling law is 3 USC Sec. 19.
If everyone on the succession list is wiped out, Minnick becomes President once he becomes Speaker.
If the wipeout happens after noon on 20 January 2009, and a Cabinet secretary on the succession list became President due to 3 USC Sec. 19(d), ey would remain President until noon 20 January 2013 despite the election of a Speaker or a President pro tem.
If the wipeout happens before noon on January 20, and a Cabinet secretary on the succession list became President due to 3 USC Sec. 19(d), ey would remain President until noon 20 January 2009. If before then, Minnick elects himself Speaker, he becomes President on noon 20 January 2009 for a full term.
In any of the above scenarios, if Obama or Biden were merely incapacitated instead of killed and then after noon on 20 January 2009 recovered, they would take office once they were capable, displacing the person who was their placeholder.