Regarding searchinput, generally its not a good practice to have other distractions on security related pages such as register, login and password recovery. However if you wish to do that there are some properties you can set on on <portal:SearchInput in layout.master including:
Breadcrumbs is a separate issue. Breadcrumbs can be used to show the relationship of a content page to its parent content page to show a user where they are in the page hierarchy. Registration and Login and other non content pages are not child pages of any content page so breadcrumbs cannot be shown there. Nor is it needed since there is always a home link visible. I do not have a way for you to show breadcrumbs on those pages.
Hope that helps,