Lock Screen Pattern Too Small After Android 4.3 Update on Samsung Galaxy S3
1) Does there exist a way to make the lock screen pattern larger on
  Android 4.3 so that it looks more like it previously in Android 4.1.2
  without the use of a third-party app?
  I don't intend on using the
  space provided for widgets above the pattern, so a solution that
  causes this functionality to be removed in the process is acceptable.


As from above images it seems that the upper widget screen is comparatively similar in both images, I think the width of lock pattern is reduced.

So, I think you can't change the size of lock pattern without the use of a third-party app


  2) If no native solution exists, is there a third-party app that I can
  download that will allow me to substitute the default home lock screen
  pattern for one that is larger? I am aware that there are app-specific
  lock screens with large patterns, but I am referring to the home lock
  screen only.


AFAIK, You can't replace the home lock screen.
You can disable the home lock screen and enable the third party app lock screen, which will work as lock screen.

Example: Go Locker!