What is the fastest/simplest way to add/manipulate PHP on a page?
You are in fact asking four questions here.


As jhedstrom said, you need to use the hook_block_info() / hook_block_view() hooks. A quick google search pointed to to this example: http://www.summasolutions.net/blogposts/how-create-basic-block-code-new-drupal-7
You can use the hook_block_view_alter() or hook_block_view_MODULE_DELTA_alter() hooks.
You can find detailed information about how to create a form in drupal in the quickstart guide. Note that this is for Drupal 6 but works the same way in Drupal 7 for the most part.
This does not require any code at all. In fact, doing this with code is doing it the wrong way. Instead, you can disable account registrations by visitors at Configuration > Account settings, set "Who can register accounts?" to Administrators only.