how can I manipulate functionality or the structure of an existing form? How can I manipulate functionality or the element's structure of an existing form in a different way and create my own template? 

For example, I can use below code to display a user login form page, but this code doesn't have any flexibility and doesn't allow me to edit or customize each part of this form element separately and I can't change the look and order of this form straightforwardly.
Imagine that I want to add show password checkbox and other visual effect and style to that's form element.

Page--user.html.twig

{{ page.content }}
You should separate functional and visual changes. 

All the functional changes should done by hook_form_alter and the rest with preprocess functions (if needed) but rather with twig.

If you use devel module, you can use the kint() function in twig, so you will see you can also filter page.content. But if that's not enough you can make an own template to this particular form, you will need twig debug enabled, which will show you HOW TO change a part of the dom (once again preprocess function and template naming convention suggestions).