How do you use output redirection in combination with here-documents and cat?

There are multiple ways to do this. The simplest is probably this:

cat <<EOF | sh
touch somefile
echo foo > somefile
EOF

Another, which is nicer syntax in my opinion:

(
cat <<EOF
touch somefile
echo foo > somefile
EOF
) | sh

This works as well, but without the subshell:

{
cat <<EOF
touch somefile
echo foo > somefile
EOF
} | sh

More variations:

cat <<EOF |
touch somefile
echo foo > somefile
EOF
  sh

Or:

{ cat | sh; } << EOF
touch somefile
echo foo > somefile
EOF

By the way, I expect the use of cat in your question is a placeholder for something else. If not, take it out, like this:

sh <<EOF
touch somefile
echo foo > somefile
EOF

Which could be simplified to this:

sh -c 'touch somefile; echo foo > somefile'

or:

sh -c 'touch somefile
echo foo > somefile'

Redirecting output instead of piping

sh >out <<EOF
touch somefile
echo foo > somefile
EOF

Using cat to get the equivalent of echo test > out:

cat >out <<EOF
test
EOF

Multiple Here Documents

( cat; echo ---; cat <&3 ) <<EOF 3<<EOF2
hi
EOF
there
EOF2

This produces the output:

hi
---
there

Here's what's going on:

The shell sees the ( ... ) and runs the enclosed commands in a subshell.
The cat and echo are simple enough. The cat <&3 says to run cat with file descriptor (fd) 0 (stdin) redirected from fd 3; in other words, cat out the input from fd 3.
Before the (...) is started, the shell sees the two here document redirects and substitutes fd 0 (<<EOF) and fd 3 (3<<EOF2) with the read-side of pipes
Once the initial command is started, the shell reads its stdin until EOF is reached and sends that to the write-side of the first pipe
Next, it does the same with EOF2 and the write-side of the second pipe

I just want to point out that using meow is just as valid as EOF.

This script will append Meow! to a file named cat:

#!/bin/sh
cat <<meow>> cat
Meow!
meow

Save as cats and make it executable with chmod +x cats, then run it with ./cats:

$ ./cats
$ cat cat
Meow!
$ ./cats
$ cat cat
Meow!
Meow!

Explanation:

cat <<meow is the here document syntax. It instructs the script to pick up the block of text that follows this line, until it encounters the string meow. The contents will then be outputted (or piped along).
>> cat pipes to the file named cat.
Using > instead of >> would overwrite the file instead of appending to it.
Meow! is the contents of the here document.
meow is the end marker for the here document. The contents are, with the use of >>, appended to cat.

Piping both to stdout and to a file:

To do the piping you request, no here document is required.

cat can not both output text and pass the text along, but tee is a perfect match for what you are asking for:

echo echo | tee tee

This will both output the string echo and write echo to a file named tee.

You could also pass the output through cat, if that is part of the requirement, either with:

echo echo | tee tee | cat </dev/stdin

Or just:

echo echo | tee tee | cat

Contents of file:

$ cat tee
echo

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