Editing registration page (register.php)
I've done this -- the copy, edit then paste html -- with the Login block and it seems to be working just fine. So. I'm assuming this should work for registration as well?
The "proper" way is to copy /concrete/single_pages/register.php to /single_pages/register.php and make your changes there.
You can do the same thing with /concrete/controllers/register.php if you need to affect something there.
You definitely want to copy files out of the core so your changes don't get overwritten when you update your c5 installation.
Kirkroberts was spot on.
Copy the concrete/single_pages/register.php
edit the themepaths file in your /config directory - it tells you how in the file,
and make sure that you definitely have at least a view.php in your theme - or create a new page called register.php in your theme which calls in the content from your single page.
Then when you are in edit mode in the website you can go to /register and add various blocks to your page etc as you would any other.
Once you get your head around it it is really flexible.
I highly recommend learning to take advantage of them, as you'll save yourself a LOT of work long-term. Leave the custom mySQL stuff for something not already implemented in the c5 core.
The documentation could be better, but it's a lot better than it was a year ago! And when you run into specific issues you can search the forum... chances are someone's already asked a similar question.