order of succession loophole?

[True Federalist (진정한 연방 주의자)]

I expect that since Secretary Gates is currently scheduled to continue on as SoD in the Obama cabinet, he'll be the designated survivor and will not show up at the inauguration.  Gates doesn't need to be reconfirmed, so he'll provide the necessary continuity of government.

In addition to that, there have been numerous examples of a cabinet Secretary staying in office a few days into the term of a new President until his nominated successor is confirmed.  Most often it's been the Secretary of State, although the ast time that happened for Secretary of State was in 1913 when Taft's Secretary of State Philander Chase Knox remained in office until March 5, 1913 when Wilson's SoS William Jennings Bryan was confirmed.  Indeed the Taft/Wilson transition appears to be the last time any of the cabinet chose to hang around an extra day or two with the succeeding president being of the opposite party (Taft's Secretary of the Treasury also stayed an extra day.)  However several of Reagan's Secretaries remained a few days until GHWB's nominees took office.

[True Federalist (진정한 연방 주의자)]

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.

[True Federalist (진정한 연방 주의자)]

If there was no cabinet in place yet, and the Pres, VP, Speaker and Pro-Tempore are killed, then then any remaining members of the House would get together, elect the Speaker who would become an interim President until the House appoints someone to finish the term...

In the case of Obama --regardless if he takes the oath, the constitution gives him power at 12:00pm on January 20th 2008.

If Obama, Bryd, Pelosi and Biden were killed minutes after then Robert Gates would  become President. He was kept away as the designated survivor and didn't need to be reconfirmed by the Senate.

As discussed here, the question is whether Rice is still SoS minutes after Obama is sworn in or not.  Does the Cabinet of the outgoing president automatically lose their jobs the second that the new president is sworn in, or are they legally still Cabinet secretaries until their replacements are sworn in?

It is traditional in a change of administration for the outgoing cabinet to resign effective noon, January 20, unless as happened with Gates, the President plans to keep them on.  Thus Gates did not need to be reconfirmed.  That also was the case for William Joseph Burns, Under Secretary of State for Political Affairs, who served as Acting Secretary of State for a day until Clinton was confirmed on the 21st. However, the succession law explicitly provides that acting secretaries are not in the order of succession.