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?

[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?