What happens if President Elect Trump dies before being inaugurated, but after President Obama's term Expires?

There is no such possibility. Obama is the president right up until Trump takes the oath of office and assumes the role.

Mike Pence becomes President.

Joe Biden does not continue after January 20. Mr. Pence would become Prez.

Mike Pence becomes President and then he is allowed to pick his own Vice-President.

If Obama dies before Friday, Bidden will be president.

If Trump dies before Friday, Pence will be president

Pence.

(Biden's term as Vice President ends the same time as Obama's term as President. If it's after Obama's term expires, then Biden is no longer Vice President.)

Mike Pence would be President Elect.

The two take place simultaneously. So trump would have to die just before saying he does take the oath, in which case they might have to go and ponder that for a while.
We don't separately the VP, his being the VP would depend on the president being president. Unfortunately its never going to happen like that. We couldn't be so lucky

Obama would remain President until a new candidate could be elected to replace him. You might recall that Pence wasn't on the ballot. That is because VP is an appointed position and none of Trumps appointments take office until he does first. If Obama resigned, Biden would succeed him until another candidate was elected. If Biden resigned Paul Ryan (speaker of the house) would succeed to become President until another candidate could be elected.

The line of succession would be Mike Pence, then if HE were unavailable (i.e. dead), House Speaker Paul Ryan would be the President... not sure who it would be if Ryan weren't available, but pretty sure it'd still be a Republican. But correct me if I'm wrong!