String Manipulation Interpreter
R, 287 286 273 269 bytes



function(C,x='',`[`=gsub,`!`=intToUtf8,`?`=utf8ToInt){for(k in el(strsplit(C,'\\|'))){B=eval(parse(t='^.'['','(?&lt;=.)&gt;$'['readLines(,1)','[@#](.+)'['"\\1"',k],,T]]));x=switch((?substr(k,1,1))%%13-2,strrep(x,B),paste0(x,B),,B['',x,f=T],!rev(?x),print(x),,!sample(?x))};x}


Try it online!


-1 thanks to @Kirill L.
-4 thanks to @Giuseppe


Unrolled code and explanation :

function(C){                                      # C is the string manipulation expression
  x = ''                                          # initialize x = ''
  tokens = el(strsplit(C,'\\|'))                  # split C by pipe '|'
  for(k in tokens){                               # for each token k
    arg2 = k
    arg2 = gsub('[@#](.+)','"\\1"',k)             # replace @X or #X with "X" (in quotes)
    arg2 = gsub('(?&lt;=.)&gt;$','"readLines(,1)"',
                 arg2,perl=T)                     # replace &gt; with readLines(,1)
    arg2 = gsub('^.','',arg2)                     # remove the first character
    B = eval(parse(t=arg2))                       # evaluate the string : this will be our 
                                                  # second argument B
    A = substr(k,1,1)                             # take the first character : 
                                                  # i.e. the main command (+,-,! etc)
    x = switch(A,                                 # switch on the main command, execute the 
            '+'=paste0(x,B),                      # corresponding expression and 
            '!'=intToUtf8(rev(utf8ToInt(x))),     # store the result into x
            '*'=strrep(x,B),                      # Note: in the actual code we switch on
            '-'=B['',x,f=T],                      # the utf8 value MOD 13-2 of the command
            '$'=intToUtf8(sample(utf8ToInt(x))),
            '&lt;'=print(x)
        )
    }
    x                                             # return x (and print it implicitly)
}