Is it true that dua and keeping one's kin close could extend one's lifespan? Inversely, could severing one's relationships be a cause for someone's early demise? Ajal'ul qada' and ajal'ul qadar? For example, if a person commits suicide, does the person die earlier than the predestined time of death?
Nothing extends or reduces one's lifespan, regardless. This is the opinion that scholars mostly agree upon that when one's time is up, it will not be delayed, as Prophet Nūh ﷺ told his people:
يَغْفِرْ لَكُم مِّن ذُنُوبِكُمْ وَيُؤَخِّرْكُمْ إِلَىٰ أَجَلٍ مُّسَمًّى إِنَّ أَجَلَ اللَّهِ إِذَا جَاءَ لَا يُؤَخَّرُ لَوْ كُنتُمْ تَعْلَمُونَ
Allah will forgive you of your sins and delay you for a specified term. Indeed, the time [set by] Allah, when it comes, will not be delayed, if you only knew.
Their good deeds would have only delayed them to a decreed time. When their time came, there was no postponement:
وَلَن يُؤَخِّرَ اللَّهُ نَفْسًا إِذَا جَاءَ أَجَلُهَا وَاللَّهُ خَبِيرٌ بِمَا تَعْمَلُونَ
But never will Allah delay a soul when its time has come. And Allah is Acquainted with what you do.
Potentially, there are two hadiths that may have caused you to ask your question.
عَنْ سَلْمَانَ قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم: لاَ يَرُدُّ الْقَضَاءَ إِلاَّ الدُّعَاءُ وَلاَ يَزِيدُ فِي الْعُمُرِ إِلاَّ الْبِرُّ
Salman narrated that the Messenger of Allah (ﷺ) said: "Nothing turns back the Decree except supplication, and nothing increases the life-span except righteousness."
This hadith is graded da'īf (weak) due to Yahya ibn ad-Durais and Abu Mawdūd, but was graded hassan (good) or sahih (authentic) by others as there is another hadith with the same words narrated through Thawban, and was also documented in Al-Mustadrak by Al-Hākim an-Nisapūri.
Scholars, however, have explained that there are decrees in the manuscripts given to the angels with their orders to carry out. Such orders are the ones that may be changed by supplication. This opinion was expressed by Majmū' al-Fatāwa 15/492 by Ibn Taymiyyah, among other scholars.
This concept may be difficult to grasp, even among the companions. When the plague of Emmaus hit in Judea, 'Umar ibn al-Khattāb, who the Muslims' leader at the time, ordered everyone to supplicate to Allah, and that those who were in the land of the plague not to leave that land while the Muslims outside not to go in. Abu 'Ubaidah ibn al-Jarrāh, who was the leader of the Muslims in that land, asked 'Umar: "Are you escaping from Allah's Decree?" 'Umar reprimanded Abu 'Ubaidah and told him: "But we escape from Allah's Decree to Allah's Decree." Az-Zamakhshari mentioned in Al-Kashshāf 'an Haqā'iq at-Tanzīl that supplication would not make their lives any longer or shorter (see Musnad al-Fārūq 2/637-638 by Ibn Kathir).
Ramla bint Abi Sufyan, the wife of the Prophet ﷺ, also known as Umm Habība, had a similar concept to the one in your question. She supplicated that she would always benefit (i.e., keep alive) her husband, her father, and her brother. The Prophet ﷺ corrected her by saying that their lives' durations had already been set and would neither be brought forward not delayed:
قَالَتْ أُمُّ حَبِيبَةَ زَوْجُ النَّبِيِّ صلى الله عليه وسلم: اللَّهُمَّ أَمْتِعْنِي بِزَوْجِي رَسُولِ اللَّهِ صلى الله عليه وسلم وَبِأَبِي أَبِي سُفْيَانَ وَبِأَخِي مُعَاوِيَةَ. قَالَ فَقَالَ النَّبِيُّ صلى الله عليه وسلم: قَدْ سَأَلْتِ اللَّهَ لآجَالٍ مَضْرُوبَةٍ وَأَيَّامٍ مَعْدُودَةٍ وَأَرْزَاقٍ مَقْسُومَةٍ لَنْ يُعَجِّلَ شَيْئًا قَبْلَ حِلِّهِ أَوْ يُؤَخِّرَ شَيْئًا عَنْ حِلِّهِ وَلَوْ كُنْتِ سَأَلْتِ اللَّهَ أَنْ يُعِيذَكِ مِنْ عَذَابٍ فِي النَّارِ أَوْ عَذَابٍ فِي الْقَبْرِ كَانَ خَيْرًا وَأَفْضَلَ
Abdullah reported that Umm Habiba, the wife of Allah's Apostle (ﷺ), said: "0 Allah, enable me to derive benefit from my husband, the Messenger of Allah (ﷺ), and from my father Abu Sufyan and from my brother Mu'awiya." Allah's Apostle (ﷺ) said: "You have asked from Allah about durations of life already set, and the length of days already allotted and the sustenances the share of which has been fixed. Allah would not do anything earlier before its due time, or He would not delay anything beyond its due time. And if you were to ask Allah to provide you refuge from the torment of the HellFire, or from the torment of the grave, it would have good in store for you and better for you also."
In Sahih Muslim bi Sharh al-Nawawi 16/213 by Al-Nawawi, he explained that neither people's lifespans nor their provisions would be extended or reduced from what had already been decreed by Allah's knowledge in al-Lawh al-Mahfūdh (the preserved tablet, Arabic: اللوح المحفوظ).
أَخْبَرَنِي أَنَسُ بْنُ مَالِكٍ، أَنَّ رَسُولَ اللَّهِ ﷺ قَالَ: مَنْ أَحَبَّ أَنْ يُبْسَطَ لَهُ فِي رِزْقِهِ وَيُنْسَأَ لَهُ فِي أَثَرِهِ فَلْيَصِلْ رَحِمَه
Anas ibn Malik reported that Allah's Messenger (ﷺ) said: "He who likes that his sustenance should be expanded and his age may be lengthened should join the tie of kinship."
In Sahih Muslim bi Sharh al-Nawawi 16/114 by Al-Nawawi, he explained that the expansion in one's sustenance or one's provisions refers to its growth and its being blessed by Allah, and similarly the lengthening of one's age is by blessing one's time, guiding one to better utilization of time in what benefits one's afterlife, and protecting one from what is wasteful. It is not by increasing the number of years; rather, increasing what occupies the pre-ordained number of years, as Allah's Decree had already been determined:
وَمَا كَانَ لِنَفْسٍ أَن تَمُوتَ إِلَّا بِإِذْنِ اللَّهِ كِتَابًا مُّؤَجَّلًاَ
And it is not [possible] for one to die except by permission of Allah at a decree determined.