1) You're Elena Burgess, aren't you? = First and last names are usually made prominent in an out-of-the-blue statement. In this case, a falling is used to indicate a newly introduced piece of information into the discourse. The question tag, which appears (as is frequently the case for tags) in a separate tone unit, has the function of a true question-type tag. The rising is typically used when the speaker is really interested in making sure about the statement preceding. It acts as a real question.
2) I see you have two jobs available for psychologists = Each of the highlighted words is important for the meaning of the statement the speaker makes. Job is arguably not made prominent as this seems to be a job interview, in which case this word is 'shared' in the context of utterance. Although the information contained in this tone unit is not strictly speaking 'new', the tone unit produced has a falling tone because statements normally take what Wells (2006: 25) calls the definitive fall.
3) Well, people say I have a lot of compassion = We have said on many occasions that 'people' is a very general word and is typically not made prominent. However, it can be said that here it is used in contrast to 'I think/I say' and is, therefore, highlighted. As to the falling-rising on 'compassionate', let's recall that speakers often use this tone in a statement when they want to sound tentative about what they say. Wells (2006:27) calls this implicational fall-rise, with an implication that has not been stated. The implication here can be stated in these terms "...people say I have a lot of compassion and I agree or and I hope you find the same". It may sound as if the truth of what Elena says is left up to the interlocutor to be discerned.
4) You're compassionate? In what way? = These two tone units, each with a fall, have a question mark but we can say more than just "They're questions". In the first one, the interviewer seems to be acknowledging the idea of compassion, as if s/he were saying "aha", "mhmm", "I got it" or something of the sort. It doesn't seem to be a typical echo question and, therefore, doesn't seem to be expressing surprise or amazement at what the other speaker has said (Wells, 2006:55). In the next tone unit, the interviewer seems to be prompting Elena to expand on "how she is compassionate", so to speak. It can safely be interpreted as a finding-out question.
5) Both my parents are psychologists, too. = This sentence is said in two tone units. In the first one, 'parents' is neither prominent nor tonic because the word was said immediately before and, therefore, there's no need for higlighting it again. It's become part of the common ground. The falling-rising may be taken as an implicational fall-rise. Analyzed syntactically, this unit coincides with the subject of that sentence. Subjects can have a tone unit of their own when “a new sentence involves a change of grammatical subject” (Wells, 2006, p. 193). Furthermore, this fall may be taken as accompanying items conveying new information.
Wells, J. (2006). English Intonation: An introduction. Cambridge: Cambridge University Press.