From what I can tell the Pentagon has managed to evade the spirit if not the wording of the Antideficiency Act; but as you can imagine no one is kicking up a ruckus about it since these are payments to war widows. A safe bet by the Pentagon - as punching war widows in the face is bipartisan suicide for a politician. Though some officer or desk general might quietly retire in a year's time if a scapegoat is needed.
As Fisher House is a charity, it is also possible that no binding contract to reimburse has been signed and so the Antideficiency Act hasn't been breached.
Update: I realise I didn't directly answer the question as worded, so an addendum:
Can the Executive Branch legally enter into a contract to repay Fisher House?
If the Executive already had a contractual agreement (fixed or
open-ended) to reimburse generic charities or Fisher House
specifically prior to the shutdown; these contracts still stand. Any
restructuring of these contractual obligations also stands providing
new debt isn't created due to the restructuring. Since governments
tend to have AAA bond ratings - the risk is about as low as cash and
so DJClayworth's answer comes into effect: Bob gives Sue cash to give
Bob => Sue "cash" isn't the same as the
Sue=>Mary cash, but as
cash is an IOU from the rest of the world; any extremely low risk IOU
is cash; and so Sue can treat Bob's IOU that way.
But it is iffy whether Bob can shift the owner of an IOU or create and
delete IOUs in this fashion. Basically Bob might not have a legal leg
to stand on if Mary still demands Bob's original payment to Mary after
the shutdown; Sue's charitable payment being neither here or there.
Of course it might be that Mary's payments from Bob were always
means-tested and so any charitable donation voids Bob's
responsibilities. Arg. This gets complicated. Why can't a government
just work properly?
Under what authority can the Executive take out loans without Congress's approval?
None or N/A depending on context. As the emergency clauses of the Antideficiency Act are
essentially Congress' approval of Executive discretion. Wrap your head
around that logic, at least in the current political climate.
If they do have this power, why can't they end the shutdown immediately (FED reserve loan)?
If they did have this power (USA on Earth-2), the current shutdown
simply "splits" into two parts:
Part A) Executive can now spend money. For this part, the shutdown is "solved".
Part B) The Congress no longer has the power of the purse. For this part, the shutdown remains unresolved until Congress reconciles with
itself or the appropriate article of the constitution is altered or