If the President, Vice President, and Speaker of the House died

[Lincoln Republican]

If the President, Vice President, and Speaker of the House died, the President Pro Tem of the Senate would become President.

However, if the House acted quickly and chose a new Speaker immediately after the previous Speaker died, could this new Speaker take their place in the line of succession and in fact become President?

[Del Tachi]

If POTUS, VPOTUS, and SOTH all died and the PPT became president, not acting president.

When the House elects a new SOTH, given that the Senate has not selected the new Vice President, than SOTH becomes the acting Vice President not the President. 

[SteveRogers]

If the President, Vice President, and Speaker of the House died, the President Pro Tem of the Senate would become President.

However, if the House acted quickly and chose a new Speaker immediately after the previous Speaker died, could this new Speaker take their place in the line of succession and in fact become President?

No. The Senate President Pro tempore would technically be Acting President, but they would remain in office and serve out the whole remainder of the term. Under the Presidential Succession Act, the election of a new Speaker would not bump a Senate PPT acting as President from office. However, either a new Speaker or PPT could bump someone further down on the line of succession from office and take their place as acting president. So if both the Speaker and the President pro tem were dead and a cabinet member became Acting President,  whichever house of congress elected a new officer first could see that person take over as Acting President for the whole term.

If POTUS, VPOTUS, and SOTH all died and the PPT became president, not acting president.

Actually the Constitution and the Presidential Succession Act make it clear that if anyone other than the VP takes over as president, they would technically be the Acting President even though they might serve out the President's entire unexpired term.

When the House elects a new SOTH, given that the Senate has not selected the new Vice President, than SOTH becomes the acting Vice President not the President.

This is wholly incorrect.  

[Oldiesfreak1854]

Then get ready for President Leahy.

[Lincoln Republican]

If the President, Vice President, and Speaker of the House died, the President Pro Tem of the Senate would become President.

However, if the House acted quickly and chose a new Speaker immediately after the previous Speaker died, could this new Speaker take their place in the line of succession and in fact become President?

No. The Senate President Pro tempore would technically be Acting President, but they would remain in office and serve out the whole remainder of the term. Under the Presidential Succession Act, the election of a new Speaker would not bump a Senate PPT acting as President from office. However, either a new Speaker or PPT could bump someone further down on the line of succession from office and take their place as acting president. So if both the Speaker and the President pro tem were dead and a cabinet member became Acting President,  whichever house of congress elected a new officer first could see that person take over as Acting President for the whole term.

If POTUS, VPOTUS, and SOTH all died and the PPT became president, not acting president.

Actually the Constitution and the Presidential Succession Act make it clear that if anyone other than the VP takes over as president, they would technically be the Acting President even though they might serve out the President's entire unexpired term.

When the House elects a new SOTH, given that the Senate has not selected the new Vice President, than SOTH becomes the acting Vice President not the President.

This is wholly incorrect.  

So, if whoever takes over as Acting President, whether it be the Speaker or the President Pro Tem, do they then nominate someone who would become Acting Vice President to take over as Acting President should the Acting President die or be otherwise precluded from performing their duties as Acting President?

[SteveRogers]

So, if whoever takes over as Acting President, whether it be the Speaker or the President Pro Tem, do they then nominate someone who would become Acting Vice President to take over as Acting President should the Acting President die or be otherwise precluded from performing their duties as Acting President?

The Acting President would nominate a new Vice President as per the 25th amendment. That person would be the actual Vice President, not just acting VP. If the Acting President died/ resigned, the VP would succeed to the presidency as usual and would be President, not just acting president.

[True Federalist (진정한 연방 주의자)]

Under the Tyler precedent, a person who acts as President and will continue to do so until the end of the current term is the President, not Acting President, and it probably would be only the snarky talk hosts of the opposing political viewpoint who would call him Acting President.

[Del Tachi]

So, if the President, Vice President, and the Speaker of the House die and the Senate PPT becomes President, who is the acting Vice President?

If the House elects a new Speaker (assuming the Senate has not confirmed a new Vice President), is he not the acting Vice President by nature of being the first in-line? 

[True Federalist (진정한 연방 주의자)]

So, if the President, Vice President, and the Speaker of the House die and the Senate PPT becomes President, who is the acting Vice President?

If the House elects a new Speaker (assuming the Senate has not confirmed a new Vice President), is he not the acting Vice President by nature of being the first in-line? 

There is no such thing as acting Vice President.  The Vice Presidency combines two functions in one.  One is the successor to the current President, who would initially be the Secretary of State in this scenario, but assuming the now traditional method of selection is followed would quickly become the newly selected President pro tem of the Senate, since there would be no need to seek the position.  That person would fulfill the Vice President' other function of President of the Senate, and continue to do so even after the House chose a Speaker to be the Presidential successor. (I assume the House would need more time to act than the Senate. If it doesn't then the new PPT would never be first in the line of succession.) The House would need to choose a Speaker (and the Senate a PPT) before Congress could approve a Vice Presidential nominee, so the position of first in the line of succession would be held by two or three different people before a new Vice President was put in office whereas the position of presiding officer of the Senate would be held by one person.

[MASHED POTATOES. VOTE!]

So, if the President, Vice President, and the Speaker of the House die and the Senate PPT becomes President, who is the acting Vice President?

If the House elects a new Speaker (assuming the Senate has not confirmed a new Vice President), is he not the acting Vice President by nature of being the first in-line? 

There is no such thing as acting Vice President. 

Indeed, though there was a proposition back in the 60s to give this title to the first person in the line of succession (by current law the Speaker) if there is no Vice President.

Also, James Eastland was sometimes refeered to as the Acting Vice President when, after Agnew's resignation and then Ford succession, he was President of the Senate, although he was not the first person in line, merely second after Speaker Carl Albert.

Still, this is a cool photo from that period:

   

[Dancing with Myself]

Thank Goodness Al Haig isn't around still.