Using variables inside a bash heredoc

Don't use quotes with <<EOF:

var=$1
sudo tee "/path/to/outfile" > /dev/null <<EOF
Some text that contains my $var
EOF

Variable expansion is the default behavior inside of here-docs. You disable that behavior by quoting the label (with single or double quotes).

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