Has any US President-Elect ever failed to take office? Either by refusing, or by some health or legal issue which has come up between Polling Day and Inauguration Day?
No. To date, every president-elect has deliberately sought the job. No one has ever been drafted into it. So refusing the presidency has been an unlikely result. That will most likely continue.
As previously mentioned, several presidents have died early in their term. But the only presidential candidate who died after the election and before the transition was Horace Greeley in 1872, and he lost. Greeley actually died before the electoral college vote. If a winning candidate died at that point, such a candidate would technically never be the president-elect.
Example source: http://presidentelect.org/art_deathpresidentelect.html
If this did happen, would the President-Elect ever technically be considered President?
No. She or he would never take the oath of office and by the Greeley precedent would not have been eligible at the time of the transition. The constitution (as currently amended) specifically provides for this case, making the vice-president-elect the president.
And what is the legal status of the President-Elect anyway? Does he/she have any legally binding power?
A president-elect has no official powers. However, custom allows the president-elect to make decisions about things like appointments that will officially occur after the transfer of power. It's also not uncommon for incumbents to ask the next president's opinion on matters that will carry over.
In the 2009 transition, the Secretaries of Defense and Homeland Security continued past the end of George W. Bush's term. The Secretary of Homeland Security was only in office for a day (to allow for the next one to be confirmed by the Senate). It's unclear what would have happened if the president-elect (Barack Obama) had died after notifying people that he wanted to retain these two and before the transition. Presumably the vice-president-elect could change or confirm these decisions prior to the transition. In absence of that, it's unclear what would happen.