In answer to your first question, there's no parameter substitution because you've put the delimiter in quotes - the bash manual says:
The format of here-documents is:
<<[-]word here-document delimiter
No parameter expansion, command substitution, arithmetic expansion, or pathname expansion is performed on word. If any characters in word are quoted, the delimiter is the result of quote removal on word, and the lines in the here-document are not expanded. If word is unquoted, all lines of the here-document are subjected to parameter expansion, command substitution, and arithmetic expansion. [...]
If you change your first example to use <<EOF
instead of << "EOF"
you'll find that it works.
In your second example, the shell invokes sudo
only with the parameter cat
, and the redirection applies to the output of sudo cat
as the original user. It'll work if you try:
sudo sh -c "cat > /path/to/outfile" <<EOT
my text...
EOT