(1) On crystal sets, is it better to use small-gauge enameled wire, or
larger gauge insulated wire for greater wrap spacing?

I don't think it's that crucial as a rule, though it would depend on a number
of variables (dimensions of coil form/s, freq. range desired, and so on). I
have generally tended to use enameled wire, anywhere from 26g. to 14g., simply
because that's what I had on hand. The larger wire you use may be working for
more gain in your set than in one with the same size coil but with smaller
wire, all things being equal (though they rarely are)... so it's a matter of
tinkering and trying it out. I find it easier to use smaller enameled wire for
winding and so on. Litz wire would be the best, though it's more difficult (and
expensive) to come by.

I wonder if the greater number of
wraps or closer spacing of the smaller wire would gain anything...?

Again, it depends on the intended freq. range and desired use of the
builder/user. Additional windings on a coil will effectively lower the freq.
range of the coil (i.e., if your current coil doesn't reach down into the 500
or 600 kHz portion of the MW band, adding windings on your coil will get you
there), and having them more or less closely spaced will affect those aspects
as well. You would benefit from such changes only if you desired what those
changes would give you, but if you're happy with what you currently have, then
it wouldn't 'gain' you anything...

(2) On the inductive secondary coil, which avoids a direct connection
to the antenna and also increases selectivity, is it better to have
that coil the same diameter as the main coil and make it fixed

Again, it really depends on what you want as an end result. A more loosely
coupled inductor will have different properties than a tightly-fixed set up,
and while this may be more flexible for some users, others may wish to keep
things simpler. I have almost always used a fixed secondary formed over the
primary, though I've played with movable pick-up coils as well. For SW work,
the movable type seemed to help with selectivity more than a fixed type, and
the addition of an antenna trap/tuner in the circuit helped even more.

OR
moveable wrt to the main coil, or is it better for that secondary coil
to be smaller so it can slide within the larger main coil? I have
used the latter method with some success, but have not tried the
first.

I don't believe I've ever attempted having the secondary coil inside the
primary coil form, so I can't really comment with any credibility (not that
this fact stopped me from commenting on the rest of this post, hahah!). I have
used a number of different set ups with respect to inductor size, windings,
secondary or 'pick-up' coil size and windings, and found that each had their
own advantages and disadvantages, and depending on what I wanted that day, I
chose accordingly.
I guess the upshot of this entire rambling post from an intermediate-knowledge
xtal set builder, is that every element of the set affects all other elements
in a fundamental way, and each and every variation or combination of those
variables results in essentially a different receiver. What is "better" to some
may be useless to others (much like antennas in general) so it's sort of a moot
point, EXCEPT that simply going for it and experimenting with all the
variations is so much fun that you'll know when you've come up with something
that really IS better for you, and keep improving from there. I may need to
build a set with an inner secondary, since I had not ever really thought about
that variation before... might result in crap, might result in some surprising
reception.
Geez, I hope this has been of some help.
And hopefully there are others in here who have a better knowledge of your
questions, and they can provide more insight.

Linus