how can I manipulate functionality or the structure of an existing form?
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).