from a directory (usr/share/doc) that contains multiple folders, zipped and unzipped, how do I copy those that are zipped (tar, tgz, gz, etc) with html files inside into another directory (directpry A)?

tar is NOT a zipped file in and of itself - it is simply a bundle of files. zip implies compression and a .tar isn't compressed at all. A .tar.gz or .tgz IS a gzipped TAR file (that is both bundled and compressed). A .gz file is compressed but may not necessarily be a bundle. A zip (which is different than gzip) is compressed and usually a bundle but a zip could contain just one file.

Your question is a little difficult to answer because there is no way to readily tell which files have html files in them without running other commands.

For example to see what was in a .tar or .tar.gz (.tgz) you could run:
tar tvf <filename> |grep -i htm

The command to see what was in a zip would be different.

You'd essentially need to write a script that:
1) Found all files that were of the types you specified.
2) Determined WHICH type (e.g. gzip or zip) the file was.
3) Ran the command to examine the contents of the file based on its type to determine if it contained htm files.
4) Do the copy of the file to new directory if it did contain such files.