Answer

1. One can omit most of it beyond the two blessings (Barukh She’amar and Yishtabach. It is preferable to add at least one more chapter in the middle, such as Ashrei). A proof of this is that someone who arrives late to synagogue may skip, and according to the Geonim (and this also seems to be the view in the Mishnah Berurah) there is no need to make it up afterward.
2. The blessings of the Shema are obligatory.
3. Rivash, responsum 412, wrote that Tachanun is only a custom. Rabbi Ovadia (Yabia Omer, vol. 3, no. 11, sec. 8) permitted someone hurrying to work to omit it. And as is well known, the Jewish people are pretty lax about this.
4. Merely a custom (Jerusalem Talmud, Ta’anit 4:5). It sounds there as though they wondered whether it may even be said at all without libations, and the conclusion is that it may be said and is certainly not obligatory. But it is noted there that such a custom exists.
5. It seems to me that, plainly speaking, this is obligatory (“so we will render the offerings of our lips”). But people are generally lax about it, and I do not know the source for that.
6. Plainly speaking, it seems that this is not obligatory but rather a custom. Only in the Amidah of the High Holy Days is it part of the formal institution of the prayer itself.

But one should know that customs too must be observed (“do not forsake”; vow), and therefore one needs reasons if one wants to be lax. Plainly speaking, annulment of vows is also required (unless the custom was based on a mistake. Here I do not think that is the case. Even if you thought this was a halakhic obligation and it turns out you were mistaken, the broader public thinks so, and I do not think there is a mistake here that uproots the vow retroactively without annulment).