Applying to graduate school at my undergraduate institution
Policies vary widely by program, but many top programs tend to encourage—and some require—that their students apply and attend other graduate schools. In part this is to encourage diversity in training experiences, so that students are not working in a single group for an extended period of time. I have heard of programs where they will admit their own students only if they haven't gotten in elsewhere (but students at such programs usually can get in somewhere if they are sufficiently qualified!).

As for the last question, again, this really does depend on the program. However, in most cases, I think the fact that you attended the same university but in a different department means you won't be viewed as one of "their" students, unless you were closely affiliated with that program somehow—for instance, if you did undergraduate research with a faculty member in that program.