David Carlile has a point in mentioning
TeX-newline-function, which is, by default, set to
newline. Looking the latter's docstring, we find:
If ‘electric-indent-mode’ is enabled, this indents the final new line
that it adds, and reindents the preceding line. To just insert
a newline, use M-x electric-indent-just-newline.
I can reproduce the behavior you describe with my settings, and it appears
electric-indent-mode is on by default (I don't have it explicitly in my init file, and it was on here). Indeed, disabling it, the editor does let me "Start of a new paragraph." without the extra spaces.
electric-indent-mode (in LaTeX-mode, or everywhere, as you prefer) or setting
electric-indent-just-newline may be two good alternatives for you.
One sensible way to disable
electric-indent-mode for specific modes is suggested at Disabling electric-indent-mode for one mode (latex mode) only. Which is to create a function to disable
electric-indent-local-mode and then hook it to the mode(s) of interest:
(defun JH/remove-electric-indent-mode ()
(add-hook 'LaTeX-mode-hook 'JH/remove-electric-indent-mode)
(add-hook 'tex-mode-hook 'JH/remove-electric-indent-mode)
;; and so on
However, you seem to be fighting auto-indent in general, when it should be there to help you. For example, the reason in the first place why "When I type the close curly braces and ENTER AUCTeX will bring the footnote back to the first column." seems to be that you set
(setq TeX-brace-indent-level 0). Furthermore, the fact that "when I type "paragraph." and hit ENTER it moves "Start" by two spaces" seems to indicate you have indentation set to two spaces. However, when you manually indented your footnote, you added four spaces.
The default behavior seems to rely heavily on auto-indent doing the right thing, which makes sense. However, I think you could get things to work quite close to what your are picturing, along with auto-indent, with:
(setq TeX-brace-indent-level 4)
Which will set your footnotes (and any other open braces) indented to four spaces, while the rest remains with two spaces. That alone should allow you to type what you describe.