Yes you can run Apache on your PC, and I presume by that you mean under Windows. http://www.apache.org/

And from I think I understand most if not all, mysqladmin, dbtools, mysqlfront can't connect to the remote because they can't so dumping is used?
No they probably can't connect to your remote servers, because they are running locally. However with phpmyadmin, you run it locally, get the dump of your database. You then login to your remote host, and presumably they will have it installed (most do, or you can install it yourself), and run the dump from there.

Would I have Apache, MySQL and etc, etc installed on my PC then build it all, check it, and then dump it onto my remote Linux?
Your only dump would be the data from your MySQL database. Your scripts you would just upload to your server using like FTP. Be sure though, that you don't use global variables that are platform dependant. Like on your computer $REMOTE_ADDR might work, but on your remote host it might default to a value of 127.0.0.1, which is local.

$REMOTE_ADDR is a bad example really, because locally it is always going to be 127.0.0.1 anyway But server side, it should not be this.