Am I right in thinking that the (?i) has the same effect as an i qualifier with xmsr?

Yes — almost. In, e.g.,  s{ ... }{...}xmsi the  /i modifier affects the regex match globally. A  (?i) extended pattern only has effect from the point of its appearance in the regex to the end of the regex pattern scope — which may or may not be the end of the regex! (See Extended Patterns. I must apologize: I did not mention that extended patterns are only available from Perl version 5.10 onward; I assume you have this.) I prefer the embedded  (?i) form because it is more visible and because it gives more control: case sensitivity can be turned on and off at will in a regex, and its scope closely controlled. In general, "my own preferred practices" are that the  qr// operator has only the  /xms modifiers applied to the operator, and all other modifiers , e.g., (?i), scoped within the operator. This practice just generalizes to the  m//xms and  s///xms operators. The latter two operators can also take  /g /e /r modifiers which can only be applied to the operator as a whole.

... none of the spaces in { ... } are treated as matchable characters? I suspect it's ... x ...

Yes, exactly. Allowing whitespace not to be part of the pattern eases the strain on these old eyes, a great blessing after so many years toiling in the scriptorium. See  /x in Modifiers.

I find your final assignment ($var = $var =~ regex =~ regex) rather confusing, as it seems to need to work from left to right at some times and from right to left at others.

It might have been helpful if I had thrown in a few disambiguating parentheses. But you can always add your own with O and B::Deparse:
(Here I wanted to give an example of the deparsed code, but for some reason I don't understand, it didn't work out! Quickly moving on...)
Anyway, by hand:

c:\@Work\Perl\monks>perl -wMstrict -le
"my $curdir   = 'Y:\Music\Schubert\Lieder\Terfel';
 my $startdir = 'Y:\mUsIc';
 ;;
 my $plsname =
   ((($curdir =~ s{ \A (?i) \Q$startdir\E \\? }{}xmsr) =~ tr{\\}{_}r) 
+. '.pls');
 print qq{'$plsname'};
"
'Schubert_Lieder_Terfel.pls'
[download]

Working in order of precedence (more or less inside-out in this case):

1. ($curdir =~ s{ \A (?i) \Q$startdir\E \\? }{}xmsr)
   A substitution is done on  $curdir and the substituted string returned courtesy of the  /r modifier of 5.14+;
2. (($curdir =~ s{ ... }{}xmsr) =~ tr{\\}{_}r)
   The string returned by the  s///r substitution is operated upon by  tr///r and the translated string is returned;
3. ((($curdir =~ s{ ... }{}xmsr) =~ tr{\\}{_}r) . '.pls')
   The string  '.pls' is appended to the string returned by  tr///r and;
4. The whole thing is finally assigned to $plsname. Whew!

Is the order irrelevant, and if not, what are the rules for the direction of evaluation?

The order is very relevant, and is discussed in Operator Precedence and Associativity in perlop: see the  =~ (binding) and  . (concatenation) and  = (assignment) operators.

Update: Layout of  <ol> above changed – improved?