This answer is based on the edition of Tehillat Hashem with English Translation published by Merkos L'inyonei Chinuch 1990. Page numbers are for this edition.
The section titled "Morning Blessings" begins on page 6. It includes:
- blessing for washing hands
- asher yatzar (the blessing for the body) and elohai neshama (for the soul)
- the 15 nisim b'chol yom (daily miracles like waking up, strength, opening the eyes of the blind, etc) and the y'hi ratzon (may it be Your will) petitions that follow
- blessing for torah study, and torah-study passages
- blessings for putting on the tallit and t'fillin
This is then followed, on page 12, with the section entitled "Shacharit - the morning prayer". This begins with mah tovu (how goodly are your tents O Jacob) and proceeds through the morning service (which I won't enumerate here).
Now, to your questions:
Yes, you are supposed to pray both sections, but you might do some at home. The blessing on washing that begins the first section is something you say immediately upon arising in the morning. The other prayers in this section, at least up to torah study, are often said at home too, though I think some say them in the synagogue too and I don't know what governs this choice. (In particular, I do not know what Chabad does.)
The tallit and t'fillin are donned in the synagogue if you are praying in a synagogue, so that and everything after it should be done with the community rather than at home. (If you can't pray with a community you're still obligated to pray so you do so at home, but that is not the preference.)
I don't see Sh'ma Yisrael in the first section you list, but I do see it early in the second, before Barchu. The Sh'ma and V'ahavta, but not the other two paragraphs after, are on page 17 amidst other passages from Tanach. This is part of a longer torah-study passage. I would guess that this reading of the Sh'ma fulfills (or helps to fulfill) the obligation of torah study, but it does not fulfill the obligation to say the Sh'ma because it is incomplete and not in its proper place for that. (Note the section on page 45 entitled "Laws pertaining to the recital of Shema".) The Sh'ma of the morning service is on page 46 and following.
As for whether any of this can be skipped (e.g. do you have to read all the torah-study passaages?), consult your rabbi. But the two sections you list in your question are not redundant; you do both.