Like most things in compression, it depends.

But DEFLATE sends the list of 288 code lengths, with all the symbols being implicit.

It also compresses that list with another layer of repetition encoding and huffman encoding.

For example, if you were encoding DEFLATE's default huffman table, you would first write it like (8, repeat 143, 9, repeat 111, 7, repeat 23, 8, repeat 7), and then you would compress that with a separate tiny huffman table.

The tiny huffman used on code lengths is itself encoded as a series of 3 bit code lengths, taking up 5-ish bytes, with the order of symbols predefined in the spec.

As long as the list of symbols is small then yes, because the symbols are a subset of the entire character list. I imagine some implementations basex encode the bytestring first then indicate the original and basex encoding instead of sending the symbol list.