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='^.'['','(?<=.)>$'['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}

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• -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('(?<=.)>$','"readLines(,1)"',
                 arg2,perl=T)                     # replace > with readLines(,1)
    arg2 = gsub('^.','',arg2)                     # remove the first character
    B = eval(parse(t=arg2))                       # evaluate the string : this will be our 
                                                  # second argument B
    A = substr(k,1,1)                             # take the first character : 
                                                  # i.e. the main command (+,-,! etc)
    x = switch(A,                                 # switch on the main command, execute the 
            '+'=paste0(x,B),                      # corresponding expression and 
            '!'=intToUtf8(rev(utf8ToInt(x))),     # store the result into x
            '*'=strrep(x,B),                      # Note: in the actual code we switch on
            '-'=B['',x,f=T],                      # the utf8 value MOD 13-2 of the command
            '$'=intToUtf8(sample(utf8ToInt(x))),
            '<'=print(x)
        )
    }
    x                                             # return x (and print it implicitly)
}

Python 2, 215 219 209 208 bytes

from random import*
I=raw_input;o=''
for t in I().split('|'):p=t[1:]=='>'and I()or t[2:];exec"o=o[::-1] o*=int(p) 0 print(o) o=''.join(sample(o,len(o))) o=o.replace(p,'') o+=p".split()[ord(t[0])*5%11]
print o

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-4 because raw_input is required.

9 bytes thanks to Embodiment of Ignorance; 1 byte from Ascii-only.

Ruby -palF\|, 146 142 bytes

r='';$F.map{|i|x=i[1]!=?>?i[2..-1]:gets.chomp;eval %w[r.reverse! r*=x.to_i 0 $><<r r=r.chars.shuffle*'' r.gsub!x,'' r+=x][i[0].ord*5%11]};$_=r

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Port of Chas Brown's Python answer. Does not print newlines after output.

As usual, Ruby 2.6 version will be 2 bytes shorter with endless range indexing (i[2..]).