FOR THE REASONS FOR HER DEATH:
(1) Her substancial military support, provided by her father (Germanicus) and paternal grandfather (Nero Drusus) who had formed strong and consistent links with the northern legions along the Rhine, Elbe and Danube rivers. This was passed onto Agrippina, combined with the appointment of Burrus as Praetorian Prefecti meant she had extensive military influence, assuring her own security. Thus there was a fear (somewhat inflamed by Senecca) that she would stage an uprising in order to regain her influence (especially after the failure collapsing boat). Consequently, Nero was fearful for his own life and for the security of the Empire, therefore her death was justified as necessary in order to maintain the Pax Romana.
(2) Poppaea Sabina advised Nero to act in this way, as she was fearful (or jealous as claimed by Tacitus) of Agrippina; a fear originating from Sabina being a cause behind the rift in the relationship between mother and son. She therefore had a direct influence in instigating the death of Agrippina.
(3) Legitimate fear that Agrippina wanted a share in the governing of the Principate. Virtues such as ambition and resourcefulness were by Roman standards considered inherently 'masculine'. This is supported by the conjugation of the latin equivalent. Therefore, a woman who was in possession of these virtues, and so desired political influence and a life beyond the domestic duties of the Matrona in the domus, was considered "an inversion of the natural order"; someone who went against what is considered proper within the confines of social expectation [although poor, the closest comparison i can think of in modern times is the stigma placed on lesbians 'butch' etc]. This warrants Tacitus' description of Agrippina's accession to power, "the whole country became transformed, complete obedience was accorded to a woman, and not a woman like Messalina... this was a rigorous, almost masculine despotism." Thus her execution was inspired by the Roman fear of ambitious and independent women, personifed by Tacitus' archetype of 'the scheming step-mother' - employed in the parallel depictions of Livia and Agrippina. Therefore Agrippina's defiance of the natural order, became so great, and her arrogance of having attained such a positon so large, that her death was necessitated in the restoration of the more traditional perception of women.
FOR HER MARRIAGES:
(1) Gnaeus Domitius Ahenobarbus [28 AD]
Foremost, this marriage (despite its downsides) provided Agrippina was a son, Lucius Domitius Ahenobarbus [Nero] through which she could rule. More importantly, Gnaeus provided her with a son born of consular rank. To understand this, one must consider Roman sociology surrounding notions of lineage and ancestory. In accordance with the Roman belief with all favourable traits were passed through the male bloodline; women such as Cornelia would always identify themselves as the daughter of Scipio Africanis rather than the mother of the Gracchi. This belief meant that despite Agrippina's inconstestable bloodline, for her son to have the right to the successsion, he would ultimately have to be born of consular rank. The Ahenobarbus' facilitated this, and thus provided Agrippina with not just an heir, but a Patrician heir; conducive to her ultimate rise to prominence concluding after the ascention of Nero.
(2) Sallustius Crispus Passenius [41 AD]
This marriage assured her of near-immeasurable amounts of inherited weath, including both estables and monetary supplied. This was crucial to her ability to rise in prominence, eg. it allowed her to pay for bribes, to make public and military donations, to hold games in her and Nero's name, to mint coins propagating her image and thus assuring provincial support, to establish the Colonia Claudia Agrippinse [Cologne] etc... Thus without the funds acquired by her marriage to Crispus, she would not have been able to rise to prominence.
(3) Claudius [49 AD]
Perhaps the most important of all marriages. This assured her the title of Augusta and all the powers it entailed. This marriage also allowed to to ride in the carpentium, to aquire a personal Praetorian Guard, to place her loyal supporters in high positons [Vitellius and Burrus] and of course, it allowed her to remove her enemies [Lollia Paulina, Calpurnia, Domitia Lepida]. This marraiage effectively completed her rise to prominence, as it put her in a position whereby she could not only secure her own basis of power, but also that of her son's.