Follow incomplete directions
Perl, 150 149 146 145 141 140 138 136 135 133 130 126 125 124

Added +7 for -F -Xn0i

An initial attempt.

Run with the map on STDIN and the directions after the -i option, e.g.

perl -F -Xn0iRL incomplete.pl
.....A
.#.###
B....C
##.#.#
D....E
##F###


Close STDIN with ^D or ^Z or whatever works on your operating system.

incomplete.pl:

%P=0;$^I=~s``{%;=!/
/;%P=map{$_|=$F[$^H=$_+=(1,@+,-1,"-@+")[$d&amp;3]]=~/(\w)|#|^$/*~!\$;{$1}}(%P)x@F}$d-=B&amp;$'^u`eg;print%


Replace the ^H by the literal control character to get the given score

Bonus question:


There is no input that results in only I
The shortest input that results in only U is RLLRRLLRLRLRRLRRLRLRLRRLLR
The longest input needed to result in a unique set is RLLRRRLRLRLLLRRLRLLLLLRRRLLRRRLLLLLLLRRLRRRR which gives B O R