Friar Broccoli wrote:

It is my understanding that about 4 billion years ago the
moon was much closer to the earth than today (correct ?).
and as a result of bumping into dust/asteroids etc. it has
gradually lost energy and therefore moved AWAY from the
earth.


No, it has gained energy from the Earth by means of tidal forces,
vaguely like 'pumping' a playground swing higher and higher by
shifting your centre of gravity in synch with its period. In return
the rotation of the Earth has been slowed, so that our day is longer
than it used to be.

My intuition tells me that as the moon looses
energy/momentum it should be less able to resist earth's
gravity, and therefore move NEARER to the earth. Why is my
intuition wrong?


If the Moon did experience a net loss of energy its orbit would
indeed tend to contract, but see above.


I live in Quebec City Canada, which is somewhere around the
47th parallel. Since that is far north of the Tropic of
Cancer (the most northerly height of the sun in late June) I
had always just assumed that the sun would ALWAYS be south
of east in the morning, and south of west in the evening.


You seem to be mixing up two different phenomena: the height of the
Sun at noon, and the displacement of sunrise and sunset from due east
& west (and from their 'nominal' solar times of 6h00 and 18h00) respectively.

Recently I noticed that that does not appear to be the case.
In late June, if I draw a line pointing to the sun first
thing in the morning, and another just before the sun goes
down, the interior angle is a lot less than 180 degrees. It
appears that this angle is about 150 degrees on the NORTH
side of my house. This makes no sense to me and is
beginning to drive me insane. I assume I am seeing the sun
north of east/west in the morning/evening, but why?


Well, it's a little hard to explain without diagrams (you could
probably find some by Googling), but let's start by picturing an
observer on the geographic equator. The great circle running from
east to west via the zenith, perpendicular to the horizon, is called
the "prime vertical", and from 0° latitude this coincides with the
celestial equator. So when the Sun is at 0° declination, on the
vernal equinox for example, it rises due east, passes straight
overhead at noon, and sets due west. As the Sun's declination
increases over the next three months, its path remains parallel to
the prime vertical but creeps northwards, so that its rising,
culminating, and setting points are all in the northern half of the sky.

Now imagine that the observer is a little north of the geographic
equator. Although the celestial equator and his prime vertical still
meet on the eastern and western horizons, the former now appears to
be tilted slightly southward (by an angle equal to his latitude), so
that on the vernal equinox the Sun, although rising and setting due
east and west, at noon culminates south of the PV. As the spring
progresses and the Sun moves northward, its culmination point
approaches the zenith at the same pace as was observed from the GE;
meanwhile the sunrise and sunset points are also shifting, but here
the tilt of the CE makes them move north faster than they did there.
To understand why, imagine looking eastward at 0600 solar time. The
Sun's north declination places it not only north of due geographic
east, but above the horizon as well, because celestial north no
longer coincides with geographical north, being elevated by an angle
equal to the latitude (the north celestial pole has the same altitude
above the north horizon). So the Sun has already risen and its path,
still parallel to the CE, is tilted; likewise at solar 18h00 it
hasn't set yet, and the day has become longer than 12 hours.

On a certain date the noon Sun will reach the zenith; from then
through the summer solstice it will spend the entire daytime in the
northern half of the sky. The further north the observer goes, the
later this date of zenith culmination, and the Tropic of Cancer is
the latitude where it falls on the summer solstice. From anywhere
north of this point the noon Sun is always in the southern half of
the sky -- but note that during the summmer it still crosses the PV
twice a day, rising and setting on the north side. Moreover the
increased latitude exaggerates the displacement, making the
inequality of day and night more pronounced. As we get into the
higher temperate latitudes the CE has become closer to the horizon
than to the PV; the summer sun rises and sets at a shallow angle, a
long way north of east and west, making for long twilights and short nights.

At the Arctic Circle an extreme situation is reached; the CE is
'depressed' so far that on the summer solstice the Sun's diurnal
circle only touches the north horizon instead of dipping below it.
From the North Pole the CE coincides with the horizon; since the
Sun's path is (as always) parallel to the CE, its altitude doesn't
change appreciably all day.

I've gone on longer than I intended, but to sum up the northward
displacement of sunrise and sunset in summer *increases* with
latitude; while in the tropics the relative length of day and night
doesn't change very much, and the path of the Sun never strays very
far from the PV, in temperate latitudes the difference between day &
night, and accordingly the difference between the lengths of the
portions of the Sun's diurnal arc lying above & below the horizon,
become quite large.

Everything I've said can be applied to the situation in winter,
_mutus mutandis_.

--
Odysseus