2) Is what you are asking - whether you can add the links to HTML, but that they are not visible on the page (or hide them on the page)? Of course it is possible, but I am not sure how much Google will like it. Anyway, you supposed to put your breadcrumbs for the visitors, not for Google - if breadcrumbs exist, Google *may* pick them up and show them in SERPs (or may not).
If you do not want breadcrumbs to be prominent to your visitors, why not put them at the bottom of the page, just above the footer? On the other hand, if you are worried about having extra links on the page, then no, it is not possible to have breadcrumbs in search but not as links in your HTML as Google will not pick them up otherwise.
But as I said - even if you do add breadcrumbs to your pages, there are no guarantees that Google will pick them up and show them in SERPs.