Understanding grep and pipes in linux

There's 2 separate issues here.

1. grep uses Basic Regular Expressions (BRE), and $ is a special character in BRE's only at the end of an expression. The consequence of this is that the 2 instances of $ in $Id$ are not equal. The first one is a normal character and the second is an anchor that matches the end of the line. To make the second $ match a literal $ you'll have to backslash escape it, i.e. $Id\$. Escaping the first $ also works: \$Id\$, and I prefer this since it looks more consistent.¹

2. There are two completely unrelated escaping/quoting mechanisms at work here: shell quoting and regex backslash quoting. The problem is many characters that regular expressions use are special to the shell as well, and on top of that the regex escape character, the backslash, is also a shell quoting character. This is why you often see messes involving double backslashes, but I do not recommend using backslashes for shell quoting regular expressions because it is not very readable.

   Instead, the simplest way to do this is to first put your entire regex inside single quotes as in 'regex'. The single quote is the strongest form of quoting the shell has, so as long as your regex does not contain single quotes, you no longer have to worry about shell quoting and can focus on pure BRE syntax.

So, applying this back to your original example, let's throw the correct regex (\$Id\$) inside single quotes. The following should do what you want:

grep '\$Id\$' my_dir/my_file

The reason \$Id\$ does not work is because after shell quote removal (the more correct way of saying shell quoting) is applied, the regex that grep sees is $Id$. As explained in (1.), this regex matches a literal $Id only at the end of a line because the first $ is literal while the second is a special anchor character.

^{¹ Note also that if you ever switch to Extended Regular Expressions (ERE), e.g. if you decided to use egrep (or grep -E), the $ character is always special. In ERE's $Id$ would never match anything because you can't have characters after the end of a line, so \$Id\$ would be the only way to go.}

First of all, the tmpfile dance can be avoided by using sed -i with GNU sed or sed -i '' with FreeBSD's (in-place replacement).

grep -R can take multiple paths on the command line, so if you are confident that none of the paths contain spaces, you can replace $(grep -l -Re "$searchstring" "$filepath") with $(grep -l -R "$searchstring" $(cat path_list)).

This will fail if any of the paths contain spaces, tabs, or any globbing character, but so will the original.

A much more robust approach uses find and just applies sed to all of the files, trusting it not to modify files with no matches (here assuming a bash shell):

# Read newline-separated path list into array from file 'path_list'
IFS=$'\n' read -d '' -r -a paths path_list

# Run sed on everything
find "${paths[@]}" \
  -exec sed -i -r -e "s/$searchstring/$replacestring/ig" '{}' ';'

But this doesn't give you any feedback on which files it's modifying.

A lengthier version that does give you the feedback:

# Read newline-separated path list into array from file 'path_list'
IFS=$'\n' read -d '' -r -a paths path_list

grep -l -R "$searchstring" "${paths[@]}" | while IFS= read -r file; do
  sed -r -i -e "s/$searchstring/$replacestring/ig" "$file"
  echo "Modified: $file"
done